MATH 55003–55103
Theory of Functions of a Real Variable I–II
Fall 2024–Spring 2025
(Problem 10) [De Morgan] If \(X\) is a set and \(\mathcal {F}\) is a family of sets, show that
(Problem 20) [De Morgan] If \(X\) is a set and \(\mathcal {F}\) is a family of sets, show that
(Problem 30) If \(f:A\to B\) is a function, and \(E\), \(F\subseteq B\), show that
(Problem 40) If \(f:A\to B\) is a function, and \(E\), \(F\subseteq B\), show that
(Problem 50) If \(f:A\to B\) is a function, and \(E\), \(F\subseteq B\), show that
(Problem 60) If \(f:A\to B\) is a function, and we define \(f(E)=\{f(e):e\in E\}\) for all \(E\subseteq A\), are the analogues to the above formulas true?
[Definition: Relation] Let \(X\) be a set. A subset \(R\) of the Cartesian product \(X\times X\) is called a relation on \(X\); we write \(xRy\) if \((x,y)\in R\).
[Definition: Reflexive relation] A relation \(R\) is reflexive if \(xRx\) for all \(x\in X\).
[Definition: Transitive relation] A relation \(R\) is transitive if \(xRy\) and \(yRz\) implies \(xRz\).
[Definition: Symmetric relation] A relation \(R\) is symmetric if \(xRy\) if and only if \(yRx\).
[Definition: Equivalence relation] A relation \(R\) is an equivalence if it is reflexive, symmetric, and transitive.
[Definition: Partial ordering] A relation \(R\) is a partial ordering if it is reflexive, transitive, and as far from symmetric as possible: if \(xRy\) and \(yRx\) then \(x=y\).
[Definition: Totally ordered] Let \(R\) be a partial ordering on a set \(X\) and let \(E\subseteq X\). We say that \(E\) is totally ordered if, for every \(x\), \(y\in E\), either \(xRy\) or \(yRx\).
[Definition: Equivalence class] The equivalence class of an element \(x\) of a set \(X\) with respect to an equivalence relation \(R\) on \(X\) is \(R_x=\{y\in X:x R y\}\), and \(X/R=\{R_x:x\in X\}\).
[Definition: Partition] Let \(X\) be a set and let \(\mathcal {P}\) be a collection of subsets of \(X\). If, for every \(x\in X\), there is exactly one \(P\in \mathcal {P}\) that satisfies \(x\in P\), we say that \(\mathcal {P}\) is a partition of \(X\).
(Problem 70) Let \(R\) be an equivalence relation. Show that \(X/R\) is a partition of \(X\).
(Problem 80) Let \(\mathcal {P}\) be a partition of \(X\). Define \(R=\bigcup _{P\in \mathcal {P}} P\times P\). Show that \(R\) is an equivalence relation and that \(xRy\) if and only if there is a \(P\in \mathcal {P}\) with \(x\), \(y\in P\).
[Definition: Choice function] Let \(\mathcal {F}\) be a nonempty family of nonempty sets and let \(X=\bigcup _{F\in \mathcal {F}} F\). (We do not require that \(\mathcal {F}\) be a partition of \(X\).) A choice function on \(\mathcal {F}\) is a function \(f:\mathcal {F}\to X\) such that \(f(F)\in F\) for each \(F\in \mathcal {F}\).
Zermelo’s axiom of choice. If \(\mathcal {F}\) is a nonempty collection of nonempty sets, then there exists a choice function on \(\mathcal {F}\).
(Problem 90) Suppose that \(f:X\to Y\) is a function from one metric space to another. Suppose that \(x\in X\) and that \(f\) is continuous at \(x\) in the sequential sense: for all sequences \(\{x_n\}_{n=1}^\infty \subset X\) that satisfy \(x_n\to x\), we have that \(f(x_n)\to f(x)\). Use the axiom of choice to show that \(f\) is continuous in the \(\varepsilon \)-\(\delta \) sense as well. Be sure to explain where you must use a choice axiom.
[Definition: Upper bound] Let \(R\) be a partial ordering on a set \(X\) and let \(E\subseteq X\). If \(x\in X\) and \(eRx\) for all \(e\in E\), then \(x\) is an upper bound on \(E\).
[Definition: Maximal element] Let \(R\) be a partial ordering on a set \(X\) and let \(x\in X\). If \(\{y\in X:xRy\}=\{x\}\), then \(x\) is said to be maximal.
[Homework 1.1] Let \(\mathcal {F}\) be a family of sets.
Zorn’s lemma. Let \(X\) be a nonempty partially ordered set. Assume that, if \(E\subseteq X\) is totally ordered, then \(E\) has an upper bound in \(X\) (not necessarily in \(E\)). Then \(X\) contains a maximal element.
[Definition: Cartesian product of a parameterized collection of sets] If \(\{E_\lambda \}_{\lambda \in \Lambda }\) is a parameterized collection of sets, then the Cartesian product \(\prod _{\lambda \in \Lambda } E_\lambda \) is defined to be the set of functions \(f\) from \(\Lambda \) to \(\bigcup _{\lambda \in \Lambda } E_\lambda \) such that \(f(\lambda )\in E_\lambda \) for all \(\lambda \in \Lambda \).
(Problem 100) When is the Cartesian product of a parameterized collection of sets equal to the set of choice functions on the family of sets? Can you modify the axiom of choice so that it is equivalent to the statement that the Cartesian product of of a parameterized collection of nonempty sets is nonempty?
[Definition: Field] A field is a set \(F\) together with two functions from \(F\times F\) to itself (denoted \(a+b\) and \(ab\) rather than \(s((a,b))\) and \(p((a,b))\)) that satisfy the axioms
\(a+b=b+a\) for all \(a\), \(b\in F\),
\(a+(b+c)=(a+b)+c\) for all \(a\), \(b\), \(c\in F\),
There is a \(0\in F\) such that \(a+0=a\) for all \(a\in F\),
If \(a\in F\) then there is a \(-a\in F\) such that \(a+(-a)=0\),
\(ab=ba\) for all \(a\), \(b\in F\),
\(a(bc)=(ab)c\) for all \(a\), \(b\), \(c\in F\),
There is a \(1\in F\) such that \(1a=a\) for all \(a\in F\),
If \(a\in F\) and \(a\neq 0\) then there is a \(1/a\in F\) such that \(a(1/a)=1\),
\(1\neq 0\),
\(a(b+c)=ab+ac\) for all \(a\), \(b\), \(c\in F\),
(Problem 110) Show from the axioms that \(0a=0\) for all \(a\in F\).
[Definition: Ordered field] A field \(F\) is an ordered field if
\(F\) is a field,
\(F\) is totally ordered with respect to some relation on \(F\) (which we will call \(\leq \)),
If \(a\geq 0\) then \((-a)\leq 0\),
\(a-b\geq 0\) if and only if \(a\geq b\),
If \(a\geq 0\) and \(b\geq 0\) then both \(a+b\geq 0\) and \(ab\geq 0\).
(Problem 120) Show that \(\Q \) and \(\R \) (as defined in undergraduate analysis/advanced calculus) are both ordered fields. If you have taken complex analysis, show that \(\C \) is not an ordered field.
[Definition: Complete ordered field] An ordered field \(F\) is complete if, whenever \(E\subset F\) is a nonempty subset with an upper bound, then \(E\) has a least upper bound.
[Definition: Absolute value] If \(x\) is an element of an ordered field, we define \(|x|=x\) if \(x\geq 0\) and \(|x|=-x\) if \(x\leq 0\).
The Triangle inequality. If \(x\) and \(y\) are real numbers (or rational numbers), then \(|x+y|\leq |x|+|y|\).
[Definition: Extended real numbers] We introduce the symbols \(\infty =+\infty \) and \(-\infty \) denoting two objects that are not in \(\R \), and let the extended real numbers be \(\R \cup \{-\infty ,\infty \}\). (The extended real numbers are not a field!)
We extend the relation \(\leq \) by stating \(-\infty \leq r\leq \infty \) for all \(r\in \R \).
We extend the operation \(+\) by writing
If \(a\), \(b\in \R \cup \{-\infty ,\infty \}\), then
(Problem 130) Show that every subset of the extended real numbers has a supremum in the extended real numbers. (A similar argument shows that every subset of the extended real numbers has an infimum in the extended real numbers.)
[Definition: Inductive set] Suppose that \(F\) is a field and that \(E\subseteq F\). Suppose furthermore that \(1\in E\) and that, if \(x\in E\), then \(x+1\in E\). Then we say that \(E\) is inductive.
[Definition: Natural numbers] If \(F\) is a field, we define the natural numbers \(\N _F\) in \(F\) as the intersection of all inductive subsets of \(F\).
(Problem 140) If \(F\) is an ordered field, show that \(1\in \N _F\) and that \(f\notin \N _F\) for all \(f<1\).
(Problem 141) If \(n\), \(m\in \N _F\), show that \(n+m\in \N _F\). Phrase your proof in terms of inductive sets rather than in terms of induction as done in undergraduate mathematics.
[Chapter 1, Problem 8] If \(F\) is an ordered field and \(n\in \N _F\), then \(\N _F\cap (n,n+1)=\emptyset \).
[Chapter 1, Problem 9] If \(F\) is an ordered field and \(n\), \(m\in \N _F\) with \(n>m\), then \(n-m\in \N _F\).
Theorem 1.1. Let \(F\) be an ordered field and let \(E\subseteq \N _F\). Suppose that \(E\neq \emptyset \). Then \(E\) has a smallest element.
(Problem 150) Prove Theorem 1.1. For bonus points, prove this in a general ordered field; the proof in your book is only valid in complete ordered fields.
Corollary. Let \(\Z _F=\{n-m:n,m\in \N _F\}\). If \(S\subseteq \Z _F\) and \(S\) is bounded above (respectively, below) then \(S\) contains a maximal (respectively, minimal) element.
The Archimedean property. An ordered field \(F\) has the Archimedean property if, for every \(a\in F\), there is a \(n\in \N _F\) with \(n>a\).
(Problem 160) Let \(F\) be a complete ordered field. Prove that \(F\) has the Archimedean property.
[Potential homework problem] Let \(F\) and \(R\) be two complete ordered fields. Show that there is a unique bijection \(\varphi :F\to R\) that satisfies
\(\varphi (a+b)=\varphi (a)+\varphi (b)\),
\(\varphi (ab)=\varphi (a)\varphi (b)\),
If \(a\leq b\) then \(\varphi (a)\leq \varphi (b)\).
Thus (up to isomorphism) there is only one complete ordered field.
[Definition: Dense] Let \(F\) be an ordered field. A subset \(S\) of \(F\) is dense if, whenever \(a\), \(b\in F\) with \(a<b\), there is a \(s\in S\) with \(a<s<b\).
Theorem 1.2. The rational numbers \(\Q \) are dense in the real numbers \(\R \).
(Problem 170) Prove Theorem 1.2.
The axiom of countable choice. Let \(X\) be a set. For each \(n\in \N \), let \(E_n\subseteq X\). Then there is a sequence \(\{x_n\}_{n=1}^\infty \) such that, for each \(n\in \N \), we have that \(x_n\in E_n\).
The axiom of dependent choice. Let \(X\) be a set and let \(R\) be a relation on \(X\). Suppose that for each \(x\in X\), there is at least one \(y\in X\) such that \(xRy\).
If \(x\in X\), then there is a sequence (a function from \(\N \) to \(X\)) such that \(x_1=x\) and such that \(x_nRx_{n+1}\) for all \(n\in \N \).
The Bolzano-Weierstrauß theorem. If \(\{x_n\}_{n=1}^\infty \) is a bounded sequence of points in \(\R \), then there is a subsequence \(\{x_{n_k}\}_{k=1}^\infty \) that converges.
(Bonus Problem 171) Use the axiom of dependent choice to prove the Bolzano-Weierstrauß theorem. Can you do this using only the axiom of countable choice?
(Bonus Problem 172) Show that the axiom of dependent choice implies the axiom of countable choice.
(Bonus Problem 180) Use Zorn’s lemma to prove the axiom of dependent choice.
(Problem 181) Choose a standard inductive proof from undergraduate analysis and rephrase it in terms of inductive sets.
[Definition: Finite] A subset of \(\N \) is finite if it is bounded above. An arbitrary set \(S\) is finite if there is a bijection \(f\) from \(S\) to a finite subset of \(\N \).
(Memory 190) If \(S\) is finite, then there is a \(m\in \N \) and a bijection \(f:S\to \{k\in \N :k\leq m\}\).
(Memory 200) If \(S\) is a set, \(T\) is a finite set, and there exists either
An injection \(g:S\to T\),
A surjection \(h:T\to S\),
then \(S\) is also finite.
[Definition: Countable] A set \(S\) is countable if there exists an injection \(g:S\to \N \).
(Memory 210) \(S\) is countable if and only if there exists a surjection \(h:\N \to S\).
(Memory 220) All finite sets are countable.
(Memory 221) If \(S\) is a set, \(T\) is a countable set, and there exists either
An injection \(g:S\to T\),
A surjection \(h:T\to S\),
then \(S\) is also countable.
[Definition: Countably infinite] A set \(S\) is countably infinite if it is countable but not finite.
(Memory 230) \(S\) is countably infinite if and only if there exists a bijection \(f:S\to \N \).
(Memory 231) The Cartesian product of finitely many countable sets is countable.
(Memory 240) \(\Q \) is countable.
(Memory 250) The countable union of countable sets is countable.
[Definition: Uncountable] A set is uncountable if it is not countable.
(Memory 260) The real numbers are uncountable. In fact, if \(I\subseteq \R \) is an interval, then either \(I=\emptyset \), \(I=\{a\}\) is a single point, or \(I\) is uncountable.
(Problem 270) Let \(S\) be a set. Let \(2^S\) (or \(P(S)\)) denote the set of all subsets of \(S\). Show that there does not exist a surjection \(f:S\to 2^S\).
[Definition: Open interval] A subset \(I\) of \(\R \) is an open interval if there exist \(a\), \(b\in [-\infty ,\infty ]\) with \(a\leq b\) such that \(I=(a,b)=\{r\in \R :a<r<b\}\).
(Problem 280) Is the empty set an open interval?
[Definition: Open set] A subset \(\mathcal {O}\) of \(\R \) is open if it is the union of open intervals.
Proposition 1.9. In \(\R \) (but not in other metric spaces), every open set may be written as a union of countably many open intervals that are pairwise-disjoint.
[Definition: Closed set] A set in \(\R \) is closed if its complement is open.
[Definition: Closure] If \(E\subset \R \), then \(\overline E=\bigcap _{F:F\text { closed}, E\subseteq F} F\).
(Memory 290) If \(E\subseteq \R \) then \(\overline E\) is closed. If \(E\subseteq \R \) is closed then \(E=\overline E\).
(Memory 300) If \(E\subseteq \R \) then \(\overline E=\{r\in \R :\)if \(\varepsilon >0\) then there is an \(e\in E\) with \(|r-e|<\varepsilon \}\).
The nested set theorem. If \(\{F_n\}_{n=1}^\infty \) is a sequence of sets such that \(F_n\supseteq F_{n+1}\), \(F_n\) is compact, and \(F_n\neq \emptyset \), then \(\bigcap _{n=1}^\infty F_n\neq \emptyset \).
[Definition: \(\sigma \)-algebra] Let \(X\) be a set and let \(\mathcal {A}\subseteq 2^X\). We say that \(\mathcal {A}\) is a \(\sigma \)-algebra of subsets of \(X\), or a \(\sigma \)-algebra over \(X\), if
(Problem 310) Show that \(2^X\) and \(\{X,\emptyset \}\) are both \(\sigma \)-algebras over \(X\).
(Problem 311) Let \(X\) and \(Y\) be sets, let \(\phi :X\to Y\) be a bijection, and let \(\mathcal {A}\) be a \(\sigma \)-algebra of subsets of \(X\). Let \(\phi (\A )=\{\phi (E):E\in \A \}\), where \(\phi (E)=\{\phi (x):x\in E\}\). Show that \(\phi (\A )\) is a \(\sigma \)-algebra of subsets of \(Y\).
(Problem 320) Suppose that \(\mathcal {A}\) is a \(\sigma \)-algebra and that \(\mathcal {B}\subseteq \mathcal {A}\) is countable. Show that \(\bigcap _{S\in \mathcal {B}}S\) is in \(\mathcal {A}\).
(Problem 321) Let \(\mathcal {G}\) be a collection of \(\sigma \)-algebras over \(X\). Let \(\mathcal {A}=\bigcap _{\mathcal {B}\in \mathcal {G}} \mathcal {B}\). Show that \(\mathcal {A}\) is also a \(\sigma \)-algebra over \(X\).
(Problem 322) Is the same true for unions of \(\sigma \)-algebras?
Proposition 1.13. Let \(\mathcal {F}\subseteq 2^X\). Let
Furthermore, \(\mathcal {F}\subseteq \mathcal {A}_{\mathcal {F}}\) and if \(\mathcal {A}\) is a \(\sigma \)-algebra with \(\mathcal {F}\subseteq \mathcal {A}\) then \(\mathcal {A}_{\mathcal {F}}\subseteq \mathcal {A}\).
We call \(\mathcal {A}_{\mathcal {F}}\) the smallest \(\sigma \)-algebra that contains \(\mathcal {F}\).
(Problem 330) Prove Proposition 1.13.
(Problem 331) Let \(X\) and \(Y\) be sets, let \(\phi :X\to Y\) be a bijection, let \(\mathcal {F}\subseteq 2^X\), and let \(\mathcal {A}\) be the smallest \(\sigma \)-algebra that contains \(\mathcal {F}\). Show that \(\phi (\A )\) is the smallest \(\sigma \)-algebra that contains \(\phi (\mathcal {F})\).
(Problem 340) Let \(\mathcal {A}\) be a \(\sigma \)-algebra on \(X\) and let \(\{E_n\}_{n=1}^\infty \subseteq \mathcal {A}\). Let
(Problem 350) Let \(\mathcal {A}\) be a \(\sigma \)-algebra on \(X\) and let \(\{E_n\}_{n=1}^\infty \subseteq \mathcal {A}\). Let
[Definition: Borel sets] The collection \(\mathcal {B}\) of Borel subsets of \(\R \) is the smallest \(\sigma \)-algebra containing all open subsets of \(\R \).
[We will assume that all students saw all material in this section in advanced calculus or real analysis.]
[We will assume that all students saw all material in this section in advanced calculus or real analysis.]
[Definition: Characteristic function] Let \(E\subset \R \) be a set. The characteristic function of \(E\) is defined to be
[Definition: Jordan content] Let \(E\subset \R \) be a bounded set. If \(\chi _E\) is Riemann integrable, then we say that \(E\) is Jordan measurable and that its Jordan content is \(\mathcal {J}(E)=\int _{-\infty }^\infty \chi _E\).
(Problem 360) Suppose that \(\{E_n\}_{n=1}^m\) is a finite collection of pairwise disjoint Jordan measurable sets. Show that \(\cup _{n=1}^m E_n\) is also Jordan measurable and that
(Problem 370) Find a bounded set \(E\) that is not Jordan measurable, but such that \(E=\bigcup _{n=1}^\infty E_n\), where \(\{E_n\}_{n=1}^\infty \) is a sequence of pairwise disjoint Jordan measurable sets.
[Definition: Measure] If \(X\) is a set and \(\M \) is a \(\sigma \)-algebra of subsets of \(X\), then we call \((X,\M )\) a measurable space. A measure on a measurable space \((X,\M )\) is a function \(\mu \) such that:
\(\mu :\mathcal {M}\to [0,\infty ]\).
\(\mu (\emptyset )=0\),
If \(\{E_k\}_{k=1}^\infty \subseteq \mathcal {M}\) and \(E_k\cap E_j=\emptyset \) for all \(j\neq k\), then
(Recall that if \(\mathcal {M}\) is a \(\sigma \)-algebra then \(\emptyset \in \mathcal {M}\).)
[Chapter 2, Problem 1] If \(\mu \) is a measure on a \(\sigma \)-algebra \(\mathcal {M}\), and if \(A\), \(B\in \M \) with \(A\subseteq B\), then \(\mu (A)\leq \mu (B)\).
[Chapter 2, Problem 2] We may replace the second condition by the condition “\(\mu (E)<\infty \) for at least one \(E\in \mathcal {M}\)”.
[Chapter 2, Problem 3] If \(\mu \) is a measure on a \(\sigma \)-algebra \(\mathcal {M}\), and if \(\{E_k\}_{k=1}^\infty \subseteq \mathcal {M}\), then
[Chapter 2, Problem 4] Let \((X,\M )\) be a measurable space. If \(E\in \M \), let \(c(E)=\infty \) if \(E\) is infinite and \(c(E)=\#E\) if \(E\) is finite. Then \(c\) is a measure on \((X,\M )\).
(Problem 380) Let \((X,\M )\) be a measurable space. There are functions from \(\M \) to \(\{0,\infty \}\) that are measures. Find two of them. (These are the trivial measures on \(\M \).)
[Definition: Length] Let \(I\subseteq \R \) be an interval, so \(I=(a,b)\), \([a,b]\), \([a,b)\), or \((a,b]\) for some \(a\), \(b\in [-\infty ,\infty ]\) with \(a\leq b\). We define \(\ell (I)=b-a\).
[Definition: Outer measure] Let \(A\subseteq \R \). The Lebesgue outer measure of \(A\) is
(Problem 390) Let \(A\subseteq B\subseteq \R \). Show that \(m^*(A)\leq m^*(B)\).
(Problem 391) Let \(S\subset \R \) be a finite set. Show that \(m^*(S)=0\).
(Problem 400) Let \(S\subset \R \) be a countably infinite set. Show that \(m^*(S)=0\). In particular, \(m^*(\Q )=0\).
Proposition 2.1. The outer measure of an interval in \(\R \) is its length.
(Problem 410) Let \(I=[a,b]\) be a closed bounded interval (that is, \(a\) and \(b\) are both real numbers). Show that \(m^*(I)=\ell (I)=b-a\).
(Problem 420) Prove Proposition 2.1.
Proposition 2.2. Outer measure is translation invariant: if \(E\subseteq \R \) and \(r\in \R \), then
(Problem 430) Prove Proposition 2.2.
(Problem 431) If \(E\subseteq \R \) and \(r\in \R \), then
Proposition 2.3. If for each \(k\in \N \) we have a set \(E_k\subseteq \R \), then
(Problem 440) Prove Proposition 2.3.
[Chapter 2, Problem 9] If \(A\), \(B\subseteq \R \) and \(m^*(A)=0\), then \(m^*(A\cup B)=m^*(A)+m^*(B)\).
[Chapter 2, Problem 10] Let \(A\), \(B\subset \R \) be bounded. Suppose that \(\inf \{|a-b|:a\in A,b\in B\}>0\). Show that \(m^*(A\cup B)=m^*(A)+m^*(B)\).
[Definition: Rationally equivalent] If \(r\), \(s\in \R \), we say that \(r\) and \(s\) are rationally equivalent if \(r-s\in \Q \).
(Problem 450) Show that rational equivalence is an equivalence relation.
(Problem 451) Let \(R\) denote rational equivalence. Let \([-1,1]/R\) be the set of equivalence classes in \([-1,1]\) under \(R\). This is a collection of (pairwise disjoint) sets.
If \(S\in [-1,1]/R\), is \(S\) finite, countably infinite, or uncountable?
(Problem 452) Is the collection of equivalence classes \([-1,1]/R\) finite, countably infinite, or uncountable?
(Problem 453) Let \(\varphi \) be a choice function on \([-1,1]/R\) and let \(V=\varphi ([-1,1]/R)\). Is \(V\) countable or uncountable?
(Problem 460) If \(q\) is rational, define \(V_q=\{v+q:v\in V\}\). Show that if \(q\), \(p\in \mathbb {Q}\cap [-2,2]\) then either \(q=p\) or \(V_q\cap V_p=\emptyset \).
(Problem 480) Show that \(m^*(V)>0\).
(Problem 490) Let \(\{q_k\}_{k=1}^\infty \) be a sequence that contains each rational number in \([-2,2]\) exactly once and contains no other numbers. Show that \(\sum _{k=1}^\infty m^*(V_{q_k}) \neq m^*\Bigl (\bigcup _{k=1}^\infty V_{q_k}\Bigr )\).
(Problem 500) Show that there exist two disjoint sets \(A\) and \(B\) such that \(m^*(A\cup B) \neq m^*(A)+m^*(B)\).
[Definition: Measurable set] Let \(E\subseteq \R \). We say that \(E\) is Lebesgue measurable (or just measurable) if, for all \(A\subseteq \R \), we have that
(Problem 501) Show that \(E\subseteq \R \) is measurable if and only if, for all \(A\subseteq \R \), we have that
(Problem 510) Show that the complement of a measurable set is measurable.
(Problem 520) Show that the empty set is measurable.
Proposition 2.10. Let \(E\subseteq \R \) be a measurable set. Let \(y\in \R \). Define \(E+y=\{e+y:e\in E\}\). Show that \(E+y\) is also measurable.
(Problem 530) Prove Proposition 2.10.
(Problem 531) Let \(E\subseteq \R \) be a measurable set. Let \(r\in \R \). Define \(rE=\{re:e\in E\}\). Show that \(rE\) is also measurable.
Proposition 2.4. Let \(E\subset \R \). Suppose that \(m^*(E)=0\). Then \(E\) is measurable.
(Problem 540) Prove Proposition 2.4.
Proposition 2.5. The union of finitely many measurable sets is measurable.
(Problem 550) Prove Proposition 2.5.
Proposition 2.6. If \(\{E_k\}_{k=1}^n\) is a collection of finitely many measurable pairwise disjoint sets, then
(Problem 560) Prove Proposition 2.6.
(Problem 561) Show that the intersection of finitely many measurable sets is measurable.
Proposition 2.13. Let \(\{E_k\}_{k=1}^\infty \) be a sequence of pairwise disjoint measurable sets. Then
[Chapter 2, Problem 26]If \(A\subseteq \R \) and \(\{E_k\}_{k=1}^\infty \) is as in Proposition 2.13, then
(Problem 570) Let \(\{E_k\}_{k=1}^\infty \) be a sequence of measurable sets. For each \(n\), let
Each \(F_n\) is measurable.
If \(m\in \N \) then \(G_m=\bigcup _{n=1}^m F_n\).
Each \(G_n\) is measurable.
If the \(E_n\)s are pairwise disjoint then \(E_n=F_n\).
If \(n< m\) then \(F_n\cap F_m=\emptyset \).
If \(n< m\) then \(G_n\subseteq G_m\).
\(\bigcup _{n=1}^\infty E_n =\bigcup _{n=1}^\infty F_n =\bigcup _{n=1}^\infty G_n\).
Proposition 2.7. The union of countably many measurable sets is measurable.
(Problem 590) Prove Proposition 2.7.
[Homework 2.1] The collection of Borel sets \(\mathcal {B}\) is the smallest \(\sigma \)-algebra that contains \(\{(-\infty ,a):a\in \mathbb {R}\}\).
[Homework 3.1] If \(A\), \(B\subseteq \R \) and \(\sup A\leq \inf B\), then \(m^*(A\cup B)=m^*(A)+m^*(B)\).
(Problem 591) If \(a\in \R \) then \((-\infty ,a)\) is measurable. (Note that we use Homework 3.1 to prove this, and so you may not use this fact to do Homework 3.1.)
Theorem 2.9. The collection \(\mathcal {M}\) of Lebesgue measurable subsets of \(\R \) is a \(\sigma \)-algebra on \(\R \) and contains the Borel sets.
(Problem 600) Show that the collection of Lebesgue measurable subsets of \(\R \) is a \(\sigma \)-algebra on \(\R \).
(Problem 610) Prove Theorem 2.9.
(Problem 611) Let \(\mathcal {M}\) denote the collection of Lebesgue measurable subsets of \(\R \). Then \((\R ,\mathcal {M})\) is a measurable space, and \(m^*\big \vert _{\mathcal {M}}\) is a measure on \((\R ,\mathcal {M})\).
[Definition: Lebesgue measure] We let \(m=m^*\big \vert _{\mathcal {M}}\) and refer to \(m\) as the Lebesgue measure.
(Problem 620) Let \(E\subseteq \R \) be measurable. Show that \(E=\bigcup _{n=1}^\infty F_n\), where each \(F_n\) is bounded and measurable and where \(F_n\cap F_m=\emptyset \) if \(n\neq m\).
[Definition: \(G_\delta \)-set] A set \(G\subseteq \R \) is a \(G_\delta \)-set if there is a sequence \(\{\mathcal {O}_n\}_{n=1}^\infty \) of countably many open sets such that \(G=\bigcap _{n=1}^\infty \mathcal {O}_n\).
[Definition: \(F_\sigma \)-set] A set \(F\subseteq \R \) is a \(F_\sigma \)-set if there is a sequence \(\{\mathcal {C}_n\}_{n=1}^\infty \) of countably many closed sets such that \(F=\bigcup _{n=1}^\infty \mathcal {C}_n\).
(Problem 630) Show that all \(F_\sigma \) sets and all \(G_\delta \) sets are measurable.
Theorem 2.11. Let \(E\subseteq \R \) and let \(E^C=\R \setminus E\). The following statements are equivalent:
Recall [Problem 510]: (v) and (vi) are equivalent.
(Problem 640) Show that (ii) and (iv) are equivalent.
(Problem 650) Show that (i) and (iii) are equivalent.
(Problem 651) Show that (v) implies (i) in the special case where \(m^*(E)<\infty \).
(Problem 660) Show that (v) implies (i) in general.
(Problem 670) Show that (iii) implies (iv).
(Problem 680) Show that (iv) implies (vi).
Theorem 2.12. Let \(E\subset \R \) be a measurable set with finite measure. Let \(\varepsilon >0\). Show that there is a collection of finitely many pairwise disjoint open intervals \(\{I_k\}_{k=1}^n\) such that
(Problem 690) Prove Theorem 2.12.
(Problem 700) Why do we need the assumption that \(E\) has finite outer measure?
(Problem 710) Give an example of a measurable set \(E\subset \R \) with finite measure and a \(\varepsilon >0\) such that there is no collection \(\{I_k\}_{k=1}^n\) of finitely many pairwise disjoint open intervals with \(\bigcup _{k=1}^n I_k\subseteq E\) and with \(m\Bigl (E\setminus \bigcup _{k=1}^n I_k\Bigr )<\varepsilon \). Note: The empty set is an open interval, and the empty collection is a finite collection of open intervals.
(Problem 720) Give an example of a measurable set \(E\subset \R \) with finite measure and a \(\varepsilon >0\) such that there is no collection \(\{I_k\}_{k=1}^n\) of finitely many pairwise disjoint open intervals with \(\bigcup _{k=1}^n I_k\supseteq E\) and with \(m^*\Bigl (\bigcup _{k=1}^n I_k\setminus E\Bigr )<\varepsilon \).
[Chapter 2, Problem 19] Suppose that \(E\) is not measurable but does have finite outer measure. Show that there is an open set \(\mathcal {O}\) containing \(E\) such that \(E\subset \mathcal {O}\) but such that \(m^*(\mathcal {O}\setminus E)\neq m^*(\mathcal {O})-m^*(E)\).
[Chapter 2, Problem 20] In the definition of measurable set, it suffices to check for sets \(A\) that are bounded open intervals; that is, \(E\subseteq \R \) is measurable if and only if, for every \(a<b\), we have that
(Memory 721) Let \(X\), \(Y\) be two metric spaces. Let \(f:X\to Y\) and let \(f_n:X\to Y\) for each \(n\in \N \). Suppose that each \(f_n\) is continuous and that \(f_n\to f\) uniformly on \(X\). Then \(f\) is continuous.
(Memory 722) A sequence of functions \(f_n:X\to Y\) is uniformly Cauchy if, for every \(\varepsilon >0\), there is a \(K\in \N \) such that if \(m\), \(n\in \N \) with \(m\geq n\geq K\), then \(d(f_n(x),f_m(x))<\varepsilon \) for all \(x\in X\). Suppose that \(\{f_n\}_{n=1}^\infty \) is uniformly Cauchy and that \(Y\) is complete. Then \(f_n\to f\) uniformly for some function \(f:X\to Y\).
(Memory 723) (The intermediate value theorem.) Suppose that \(a<b\) and that \(f:[a,b]\to \R \) is continuous. If \(f(a)\leq y\leq f(b)\), then there is an \(x\in [a,b]\) such that \(f(x)=y\).
(Memory 724) (Definition of interval) A set \(I\subseteq \R \) is an interval if, whenever \(a<b<c\) and \(a\), \(c\in I\), we also have that \(b\in I\). Then \(\{I\subseteq \R :I\) is an interval\(\}\) is the union of the following ten collections of sets:
\(\{\emptyset \}\)
\(\{\R \}\)
\(\{(a,b):a<b\) and \(a\), \(b\in \R \}\)
\(\{[a,b:a<b\) and \(a\), \(b\in \R \}\)
\(\{[a,b):a<b\) and \(a\), \(b\in \R \}\)
\(\{(a,b]:a<b\) and \(a\), \(b\in \R \}\)
\(\{(a,\infty ):a\in \R \}\)
\(\{[a,\infty ):a\in \R \}\)
\(\{(-\infty ,b):b\in \R \}\)
\(\{(-\infty ,b]:b\in \R \}\)
[Definition: Relatively open] Let \((X,d)\) be a metric space and let \(Y\subset X\). Then \((Y,d\big \vert _{Y\times Y})\) is also a metric space. If \(G\subseteq Y\) is open in \((Y,d\big \vert _{Y\times Y})\), then we say that \(G\) is relatively open in \(Y\).
(Memory 725) If \((X,d)\) is a metric space and \(G\subseteq Y\subseteq X\), then \(G\) is relatively open in \(Y\) if and only if \(G=Y\cap U\) for some \(U\subseteq X\) that is open in \((X,d)\).
[Definition: Connected] Let \((X,d)\) be a metric space. We say that \((X,d)\) is disconnected if there exist two sets \(A\), \(B\subseteq X\) such that
\(A\) and \(B\) are both open in \((X,d)\),
\(X=A\cup B\),
\(\emptyset =A\cap B\),
\(A\neq \emptyset \neq B\).
If no such \(A\) and \(B\) exist then we say that \((X,d)\) is connected.
(Memory 726) Let \((X,d)\) be a metric space and let \(Y\subseteq X\). Then the metric space \((Y,d\big \vert _{Y\times Y})\) is disconnected if and only if there exist two sets \(A\), \(B\subseteq X\) such that
\(A\) and \(B\) are both open in \((X,d)\),
\(Y\subseteq A\cup B\),
\(\emptyset =A\cap B\),
\(A\cap Y\neq \emptyset \neq B\cap Y\).
(Memory 727) A subset of \(\R \) is connected if and only if it is an interval.
(Memory 728) If \((X,d)\) is a connected metric space and \(f:X\to Y\) is continuous, then \(f(X)\) is also connected.
(Memory 729) By Problem 400 and Proposition 2.4, if \(E\subset \R \) is countable, then \(E\) is measurable and has measure zero.
Proposition 2.19. There is a set of measure zero that is uncountable.
(Problem 730) In this problem we begin the construction of an uncountable set of measure zero. We define the points \(a_{k,n}\) and \(b_{k,n}\), for \(n\in \N _0\) and for \(1\leq k\leq 2^n\), as follows.
Show that:
(Problem 731) If \(n\in \N _0\) and \(1\leq k\leq 2^n\), let \(F_{k,n}=[a_{k,n},b_{k,n}]\) be the closed interval of length \(3^{-n}\) with endpoints at \(a_{k,n}\) and \(b_{k,n}\). Show that \(F_{j,n}\cap F_{k,n}=\emptyset \) if \(j\neq k\).
(Problem 740) Define \(F_n=\bigcup _{k=1}^{2^n} [a_{k,n},b_{k,n}]=\bigcup _{k=1}^{2^n} F_{k,n}\). Show that \(F_{n}\subseteq F_{n-1}\) for all \(n\in \N \), and that \(F_n\setminus F_{n-1}\) may be written as the union of \(2^{n-1}\) open intervals each of length \(3^{-n}\).
(Problem 750) Let \(C=\bigcap _{n=0}^\infty F_n\). The set \(C\) is called the Cantor set. Show that \(m(F_n)=(2/3)^n\) for all \(n\in \N _0\) and that \(m(C)=0\).
[Homework 3.1b] If \(E\subseteq \R \) and \(m^*(E)<\infty \), and if we define \(f\) by \(f(x)=m(E\cap (-\infty ,x))\), then \(f:\R \to \R \) is continuous.
(Problem 751) Let \(\Lambda _k(x)=\frac {m(F_k\cap (-\infty ,x))}{m(F_k)}\). Then \(\Lambda _k\) is continuous and nondecreasing. Sketch the graphs of \(\Lambda _0\), \(\Lambda _1\), and \(\Lambda _2\).
(Problem 760) Suppose that \(x\notin F_n\). Show that \(\Lambda (x)=2^{-n}|\{k\in \{1,2,\dots ,2^n\}:b_{k,n}<x\}|\).
(Problem 770) Show that \(\Lambda _n(\R \setminus F_n)=\{i2^{-n}:0\leq i\leq 2^n,i\in \Z \}\) and that, if \(m\geq n\), then \(\Lambda _n(x)=\Lambda _m(x)\) for all \(x\notin F_n\).
(Problem 780) Show that \(\{\Lambda _k\}_{k=1}^\infty \) is uniformly Cauchy.
[Definition: The Cantor function] Let \(\Lambda (x)=\lim _{k\to \infty } \Lambda _k(x)\).
(Problem 790) Show that \(\Lambda \) exists and is continuous, nondecreasing, and surjective \(\Lambda :[0,1]\to [0,1]\).
[Definition: Almost everywhere] Suppose that \(E\subseteq \R \) is a set. If a property \(P\) is true for every \(x\in E\setminus E_0\), where \(m^*(E_0)=0\), we say that \(P\) is true almost everywhere on \(E\).
(Problem 800) Show that \(\Lambda '(x)=0\) for every \(x\in \R \setminus C\), and thus for almost every \(x\in \R \).
(Problem 810) Show that \(\Lambda ([0,1]\setminus C)\) is countable.
(Problem 820) Show that \(C\) is uncountable.
Theorem 2.17. If \(E\subseteq \R \) has positive outer measure, then there is an \(A\subseteq E\) that is not measurable.
(Problem 830) In this problem we begin the proof of Theorem 2.17. Let \(F\subset [-1,1]\) and let \(V_{q_k}\) be as in Problem 490. Show that either \(F\cap V_{q_k}\) is not measurable or \(m^*(F\cap V_{q_k})=0\).
(Problem 840) Suppose that \(m^*(F)>0\). Show that we must have that \(m^*(F\cap V_{q_k})>0\) for at least one value of \(k\).
(Problem 841) Prove Theorem 2.17.
Proposition 2.22. There is a measurable set that is not a Borel set.
(Problem 850) In this problem we begin the proof of Proposition 2.22. Let \(\Lambda \) be the Cantor-Lebesgue function and let \(f(x)=x+\Lambda (x)\). Show that \(f\) is continuous, strictly increasing, and surjective \(\R \to \R \).
[Chapter 2, Problem 47] If \(f:\R \to \R \) is continuous and strictly increasing, and if \(B\) is a Borel set, then \(f(B)\) is also a Borel set.
(Problem 860) Show that \(f(C)\) has positive measure.
(Problem 870) Prove Proposition 2.22.
(Problem 880) Prove part (ii) of Theorem 2.15 without using part (i).
(Problem 890) Find a sequence \(\{B_k\}_{k=1}^\infty \) such that \(B_k\) is measurable and \(B_k\supseteq B_{k+1}\) for all \(k\in \N \), but such that
The Borel-Cantelli Lemma. Let \(\{E_k\}_{k=1}^\infty \) be a sequence of measurable sets. Suppose that \(\sum _{k=1}^\infty m(E_k)<\infty \). Then \(|\{k\in \N :x\in E_k\}|<\infty \) for almost every \(x\in \R \).
(Problem 900) Prove the Borel-Cantelli lemma. Start by writing a formula for the set of \(x\in \R \) such that \(|\{k\in \N :x\in E_k\}|=\infty \) using unions and intersections. Explain carefully why your formula is true.
[Definition: Measurable function] Let \(E\subseteq \R \) be measurable and let \(f:E\to [-\infty ,\infty ]\). Suppose that for every \(c\in \R \) the set
Proposition 3.3. Let \(E\subseteq \R \) be measurable and let \(f:E\to \R \) be continuous. Then \(f\) is measurable.
(Problem 910) Prove Proposition 3.3.
(Problem 920) Let \(f:\R \to [-\infty ,\infty ]\). Suppose that \(\lim _{y\to x} f(y)=f(x)\) for all \(x\in \R \). Show that \(f\) is measurable.
[Chapter 3, Problem 24] A monotonic function defined on a measurable set is measurable.
Proposition 3.1. Let \(E\subseteq \R \) be measurable and let \(f:E\to [-\infty ,\infty ]\). The following statements are equivalent.
Furthermore, if any of these conditions is true, then \(f^{-1}(\{c\})\) is measurable for all \(c\in [-\infty ,\infty ]\).
(Problem 930) Prove Proposition 3.1.
[Chapter 3, Problem 4] If \(f^{-1}(\{c\})\) is measurable for all \(c\in [-\infty ,\infty ]\), is it necessarily the case that \(f\) is measurable?
(Problem 931) Let \(E\subseteq \R \) be measurable and let \(f:E\to [-\infty ,\infty ]\). Suppose that, for all \(c\in \R \), the set \(\{x\in E:c<f(x)<\infty \}=f^{-1}((c,\infty ))\) is measurable. Is \(f\) necessarily measurable? If not, what additional assumptions must be imposed to show that \(f\) is measurable?
(Problem 932) Let \(\mathcal {A}\) be a \(\sigma \)-algebra over a set \(X\), let \(Y\in \mathcal {A}\), and define \(\mathcal {S}=\{S\cap Y:S\in \mathcal {A}\}\). Show that \(\mathcal {S}\) is a \(\sigma \)-algebra over \(Y\).
(Problem 940) Let \((X,\mathcal {A})\) be a measurable space (that is, \(\mathcal {A}\) is a \(\sigma \)-algebra over \(X\)). Let \(f:X\to Y\) be a function and let \(\mathcal {F}=\{S\subseteq Y:f^{-1}(S)\in \mathcal {A}\}\). Show that \(\mathcal {F}\) is a \(\sigma \)-algebra over \(Y\).
[Homework 2.1] The collection \(\mathcal {B}\) of Borel sets is the smallest \(\sigma \)-algebra containing \(\{(-\infty ,a):a\in \R \).
Proposition 3.2. If \(f\) is measurable, then \(f^{-1}(\mathcal {O})\) is measurable for all open sets \(\mathcal {O}\).
(Problem 950) Prove that in fact, if \(f\) is measurable, then \(f^{-1}(B)\) is measurable for all Borel sets \(B\).
(Problem 960) If \(g\) is measurable, is it true that \(g^{-1}(E)\) is measurable for all measurable sets \(E\)?
(Problem 970) If \(h\) is measurable, is it true that \(h^{-1}(B)\) is Borel for all Borel sets \(B\)?
Proposition 3.5. Let \(E\subseteq \R \) be measurable and let \(f:E\to [-\infty ,\infty ]\).
(Problem 971) Prove Proposition 3.5, part (ii).
(Problem 980) Prove Proposition 3.5, part (i).
(Problem 981) Let \(E\subseteq \R \). Show that \(E\) is measurable if and only if the characteristic function \(\chi _E\) is measurable.
[Chapter 3, Problem 6] Let \(E\subseteq \R \) be measurable. Let \(f:E\to [-\infty ,\infty ]\). Show that \(f\) is measurable on \(E\) if and only if the function
(Problem 982) Did we need the condition that \(E\) was measurable?
Theorem 3.6. Let \(E\subseteq \R \) be measureable, and let \(f\), \(g:E\to [-\infty ,\infty ]\) be measurable functions that are finite almost everywhere in \(E\)
If \(\alpha \), \(\beta \in \R \), then \(fg\) and \(\alpha f+\beta g\) are defined almost everywhere on \(E\) and are measurable on \(E\) in the sense of Proposition 3.5, that is, in the sense that any of the extensions of \(fg\) and \(\alpha f+\beta g\) to \(E\) are measurable.
(Problem 983) If \(f\) is measurable and \(\alpha \in \R \), then \(\alpha f\) is measurable.
(Problem 990) Suppose that \(f\) and \(g\) are measurable and finite almost everywhere. Show that \(f+g\) is measurable.
(Problem 1000) Suppose that \(f\) is measurable. Show that \(f^2\) is measurable. Then prove Proposition 3.5.
(Problem 1010) Give an example a measurable function \(h\) and a continuous function \(g\) such that \(h\circ g\) is not measurable.
Proposition 3.7. If \(D\), \(E\subseteq \R \) are measurable, if \(h:E\to D\) is measurable, and if \(g:D\to \R \) is continuous, then \(g\circ h\) is measurable.
[Chapter 3, Problem 8iv] More generally, if \(E\subseteq \R \) is measurable, if \(D\subseteq \R \) is a Borel set, if \(h:E\to D\) is measurable, and if \(g:D\to \R \) is such that \(\{x\in D:g(x)>c\}\) is Borel for all \(c\in \R \), then \(g\circ h\) is measurable.
(Problem 1020) Prove Proposition 3.7.
(Problem 1021) If \(f\) is measurable, show that \(|f|\) is measurable.
Proposition 3.8. Let \(E\subseteq \R \) be measurable and let \(f_1\), \(f_2,\dots ,f_k:E\to [-\infty ,\infty ]\) be finitely many measurable functions. Then \(f(x)=\max \{f_1(x),\dots ,f_k(x)\}\) is also measurable.
(Problem 1030) Prove Proposition 3.8.
[Definition: Pointwise convergence] Let \(E\) be a set and let \(f_n\), \(f:E\to [-\infty ,\infty ]\). If \(f_n(x)\to f(x)\) for all \(x\in E\), then we say that \(f_n\to f\) pointwise on \(E\).
[Definition: Almost everywhere convergence] Let \(E\subseteq \R \) be a set and let \(f_n\), \(f:E\to [-\infty ,\infty ]\). If \(f_n(x)\to f(x)\) for all \(x\in E\setminus D\), where \(m(D)=0\), then we say that \(f_n\to f\) almost everywhere on \(E\).
[Definition: Uniform convergence] Let \(E\subseteq \R \) be a set and let \(f_n\), \(f:E\to \R \). Suppose that for all \(\varepsilon >0\) there is a \(m\in \N \) such that, if \(n\geq m\), then \(|f_n(x)-f(x)|<\varepsilon \). Then we say that \(f_n\to f\) uniformly on \(E\).
(Problem 1031) Show that uniform convergence implies pointwise convergence and that pointwise convergence implies almost everywhere convergence.
(Problem 1032) Show that none of the reverse implications hold.
Proposition 3.9. Let \(\{f_n\}_{n=1}^\infty \) be a sequence of measurable functions on a measurable set \(E\). Suppose that \(f_n\to f\) almost everywhere on \(E\) for some \(f:E\to [-\infty ,\infty ]\). Then \(f\) is measurable.
(Problem 1040) Prove Proposition 3.9 by showing that \(\{x\in E:c\leq f(x)\}\) is measurable for all \(c\in \R \). Be sure to explain why your proof works even if \(f_n\), \(f\) are allowed to be infinite.
[Definition: Characteristic function] If \(A\subseteq \R \), then the characteristic function of \(A\), denoted \(\chi _A\), is defined by
[Definition: Simple function] A function \(\varphi \) is simple if its domain \(E\subseteq \R \) is measurable, if \(\varphi \) is measurable on \(E\), and if \(\{\varphi (x):x\in E\}\) is a set of of finitely many real numbers.
[Chapter 3, Problem 6] If \(E\) is measurable and \(f:E\to \R \), then \(f\) is measurable on \(E\) if and only if
(Problem 1041) A function \(\varphi \) with domain \(E\subseteq \R \) is simple if and only if \(\varphi =\psi \big \vert _E\) for some simple function \(\psi :\R \to \R \). Furthermore, we may require that \(\psi (x)=0\) for all \(x\notin E\).
(Problem 1042) The set of simple functions contains all of the characteristic functions and is closed under taking finite linear combinations.
(Problem 1050) Suppose that \(\varphi :\R \to \R \) is simple. Show that there is a unique list of numbers \(c_1<c_2<\dots <c_n\) and a unique list of nonempty measurable sets \(E_1\), \(E_2,\dots ,E_n\) such that \(\varphi =\sum _{j=1}^n c_j \chi _{E_j}\).
Furthermore, show that \(\R =\bigcup _{j=1}^n E_j\) and \(E_j\cap E_k=\emptyset \) for all \(j\neq k\).
(Problem 1051) If \(\varphi \) and \(\psi \) are simple functions with the same domain, show that \(\max (\varphi ,\psi )\) is also simple.
The simple approximation lemma. Let \(f:E\to \R \) be measurable and bounded. Let \(\varepsilon >0\). Then there are two simple functions \(\varphi _\varepsilon \) and \(\psi _\varepsilon \) with
(Problem 1060) Prove the simple approximation lemma.
The simple approximation theorem. Let \(f:E\to [-\infty ,\infty ]\). Then \(f\) is measurable if and only if there is a sequence \(\{\varphi _n\}_{n=1}^\infty \) such that
(Problem 1061) Prove the easy direction; that is, suppose that such a sequence \(\{\varphi _n\}_{n=1}^\infty \) exists and show that \(f\) is measurable.
(Problem 1070) Prove the simple approximation theorem.
(Problem 1080) Let \(\varphi :\R \to \R \) be a simple function and let \(c_j\), \(E_j\) be as in Problem 1050. What do you expect \(\int _\R \varphi \,dm\) to equal?
(Memory 1081) (Tietze’s Extension Theorem in \(\R \)). Let \(F\subseteq \R \) be closed and let \(f:F\to \R \) be continuous. Then there is a function \(g:\R \to \R \) that is continuous on all of \(\R \) and satisfies \(g=f\) on \(F\).
(Memory 1082) Let \(F\) and \(D\) be two disjoint closed sets and let \(f:F\cup D\to \R \) be a function. Suppose that \(f\big \vert _F\) and \(f\big \vert _D\) are continuous on \(F\) and \(D\), respectively. Then \(f\) is continuous on \(F\cup D\).
Egoroff’s theorem. Let \(E\subseteq \R \) be measurable with \(m(E)<\infty \). Let \(\{f_n\}_{n=1}^\infty \) be a sequence of measurable functions on \(E\) that converges pointwise almost everywhere to some function \(f\) that is finite almost everywhere.
Then for every \(\varepsilon >0\) there is a closed set \(F\subseteq E\) with \(m(E\setminus F)<\varepsilon \) and such that \(f_n\to f\) uniformly on \(F\).
Lemma 3.10. Under the conditions of Egoroff’s theorem, if \(\mu >0\) and \(\delta >0\), then there is a measurable set \(A\subseteq E\) and a \(k\in \N \) such that \(m(E\setminus A)<\delta \) and such that \(|f_n(x)-f(x)|<\mu \) for all \(x\in A\) and all \(n\geq k\).
(Problem 1090) Prove Lemma 3.10. (Note that we will use Lemma 3.10 to prove Egoroff’s theorem, and so you may not use Egoroff’s theorem to prove Lemma 3.10.)
(Problem 1100) Use Lemma 3.10 to prove Egoroff’s theorem.
(Problem 1110) Give an example of a sequence of measurable functions on an unbounded measurable set \(E\) that converges pointwise almost everywhere to some function \(f\) that is finite almost everywhere, but such that the conclusion of Egoroff’s theorem fails.
[Chapter 3, Problem 16] Let \(I\subseteq \mathbb {R}\) be a closed, bounded interval and let \(E\subseteq I\) be measurable. Show that, for each \(\varepsilon >0\), there exists a step function \(h:I\to \R \) and a measurable set \(F\subseteq I\) such that \(h=\chi _E\) on \(F\) and such that \(m(I\setminus F)<\varepsilon \).
Proposition 3.11. Let \(\varphi :\R \to \R \) be a simple function and let \(\varepsilon >0\). Then there is a continuous function \(g:\R \to \R \) such that \(m^*(\{x\in \R :g(x)\neq \varphi (x)\})<\varepsilon \).
(Problem 1120) Prove Proposition 3.11.
Lusin’s theorem. Let \(E\subseteq \R \) be a measurable set and let \(f:E\to [-\infty ,\infty ]\) be measurable and finite almost everywhere. If \(\varepsilon >0\), then there is a continuous function \(g:\R \to \R \) and a closed set \(F\subseteq E\) such that \(f=g\) on \(F\) and such that \(m(E\setminus F)<\varepsilon \).
(Problem 1130) Prove Lusin’s theorem in the case \(m(E)<\infty \).
(Problem 1131) Prove Lusin’s theorem.
[Definition: Step function] If \([a,b]\subset \R \) is a closed and bounded interval, we say that \(\varphi :[a,b]\to \R \) is a step function if there are finitely many points \(a=x_0<x_1<\dots <x_n=b\) such that \(\varphi \) is constant on each of the intervals \((x_{k-1},x_k)\) for all \(1\leq k\leq n\).
(Problem 1132) \(\varphi :[a,b]\to \R \) is a step function if and only if there is a finite integer \(n\), finitely many (possibly degenerate!) closed intervals \(I_1\), \(I_2,\dots ,I_n\) with each \(I_k\subseteq [a,b]\), and finitely many real numbers \(a_k\) such that \(\varphi =\sum _{k=1}^n a_k\chi _{I_k}\).
[Definition: Integral of a step function] If \(\varphi :[a,b]\to \R \) is a step function and \(x_0\), \(x_1,\dots ,x_n\) are the numbers in the definition of step function, we define
[Definition: Riemann integrable] Let \([a,b]\subset \R \) be a closed and bounded interval and let \(f:[a,b]\to \R \) be bounded. We say that \(f\) is Riemann integrable on \([a,b]\) if
[Definition: Integral of a simple function] Let \(E\subseteq \R \) be measurable with \(m(E)<\infty \) and let \(\varphi :E\to \R \) be simple. Let \(\varphi (E)=\{c_1,c_2,\dots ,c_n\}\); as in Problem 1050, \(\varphi =\sum _{k=1}^n c_k \,\chi _{E_k}\), where \(E_k=\varphi ^{-1}(\{c_k\})\). We define
(Problem 1133) Let \(\varphi :\R \to \R \) be simple. If \(E\subseteq \R \) is measurable with \(m(E)<\infty \), we define \(\int _E \varphi :\int _E (\varphi \big \vert _E)\) where \(\varphi \big \vert _E\) denotes the restriction of \(\varphi \) to \(E\). If \(\varphi =\sum _{k=1}^n c_k \,\chi _{E_k}\), show that
(Problem 1134) Let \(E\) be a measurable set. Let \(\{D_1,\dots ,D_\ell \}\) be a partition of \(E\): \(E=\bigcup _{j=1}^\ell D_j\) and \(D_j\cap D_k=\emptyset \) if \(j\neq k\). Suppose furthermore that each \(D_j\) is measurable. Let \(\varphi :E\to \R \) and suppose that \(\varphi \) is constant on each \(D_j\). Then \(\varphi \) takes on at most \(\ell \) values, so is simple. Let \(b_j\) be such that \(\varphi (x)=b_j\) for all \(x\in D_j\). Show that
(Problem 1140) Suppose that \(E=[a,b]\) and that \(\varphi \) is a step function. Show that \(\varphi \) is also a simple function and that \(\int _E \varphi =\int _a^b \varphi \).
Lemma 4.1. If \(E_1\), \(E_2,\dots ,E_n\) are measurable, \(c_1\), \(c_2,\dots ,c_n\in \R \), \(\varphi =\sum _{k=1}^n c_k \,\chi _{E_k}\), and \(\bigcup _{k=1}^n E_k\subseteq E\) for some measurable set \(E\), then \(\int _E \varphi =\sum _{k=1}^n c_k\,m(E_k)\) even if the \(E_k\)s and \(c_k\)s are not as in Problem 1050.
(Problem 1150) Prove Lemma 4.1.
Proposition 4.2. Let \(\varphi \), \(\psi \) be simple functions defined on a set of finite measure \(E\).
(Problem 1160) Prove Proposition 4.2, part (i).
(Problem 1170) Prove Proposition 4.2, part (ii).
[Definition: Integral of a bounded function over a bounded set] Let \(E\subset \R \) be measurable with \(m(E)<\infty \) and let \(f:E\to [-M,M]\) be a bounded function. We say that \(f\) is Lebesgue integrable over \(E\) if
(Problem 1171) Let \(E\subset \R \) be measurable with \(m(E)<\infty \) and let \(\theta :E\to \R \) be simple. Show that
Thus all simple functions with domains of bounded measure are integrable and there is no ambiguity in using \(\int _E \theta \) to denote both the integral of a simple function and of an arbitrary Lebesgue integrable function.
Theorem 4.3. If \(f\) is Riemann integrable on \([a,b]\), then \(f\) is Lebesgue integrable over \([a,b]\) and \(\int _a^b f=\int _{[a,b]}f\).
(Problem 1180) Prove Theorem 4.3 and give an example of a bounded measurable function defined on an interval \([a,b]\) that is Lebesgue integrable over \([a,b]\) but is not Riemann integrable.
Theorem 4.4. Let \(E\subset \R \) be measurable with \(m(E)<\infty \) and let \(f:E\to [-M,M]\) be bounded and measurable. Then \(f\) is Lebesgue integrable.
(Problem 1190) Prove Theorem 4.4.
Theorem 5.7. Let \(E\subset \R \) be measurable with \(m(E)<\infty \) and let \(f:E\to [-M,M]\) be bounded. Then \(f\) is measurable if and only if it is Lebesgue integrable. (You may not use this result until we prove it in Chapter 5, but you may find it interesting at this point.)
Theorem 4.5. Let \(f\) and \(g\) be bounded measurable functions defined on a set of finite measure \(E\).
(Problem 1200) Prove Theorem 4.5, part (i).
(Problem 1210) Prove Theorem 4.5, part (ii).
Corollary 4.6. If \(A\) and \(B\) are two disjoint measurable sets of finite measure and \(f:A\cup B\to \R \) is bounded and measurable, then \(\int _{A\cup B} f=\int _A f+\int _B f\).
(Problem 1220) Prove Corollary 4.6.
Corollary 4.7. If \(E\subset \R \) is measurable and has finite measure, and if \(f:E\to \R \) is bounded and measurable, then
Proposition 4.8. If \(E\subset \R \) is measurable and has finite measure, if \(f_n:E\to \R \) is bounded and measurable for each \(n\), and if \(f_n\to f\) uniformly on \(E\), then
(Problem 1230) Prove Proposition 4.8.
(Problem 1240) Give an example of a sequence of measurable functions \(\{f_n\}_{n=1}^\infty \), each of which is bounded, defined on a common measurable domain \(E\) of finite measure, such that \(f_n\to f\) pointwise on \(E\) for some bounded measurable function \(f:E\to \R \), but such that
The bounded convergence theorem. If \(E\subset \R \) is measurable and has finite measure, if \(f_n:E\to \R \) is measurable for each \(n\), if there is a \(M\) such that \(|f_n(x)|<M\) for all \(x\in E\) and all \(n\in \N \), and if \(f_n\to f\) pointwise on \(E\), then
(Problem 1250) Prove the Bounded Convergence Theorem. Hint: Use Egoroff’s theorem.
[Definition: Finite support] Let \(E\subseteq \R \) be measurable and let \(h:E\to \R \). Suppose that there is a measurable set \(E_0\subseteq E\) with \(m(E_0)<\infty \) and such that \(h(x)=0\) for all \(x\in E\setminus E_0\). Then we say that \(h\) has finite support; if \(h\) is also bounded then we define \(\int _E h=\int _{E_0} h\).
[Definition: Integral of a nonnegative function] Let \(E\subseteq \R \) be measurable and let \(f:E\to [0,\infty ]\) be measurable. We define
(Problem 1251) Show that if \(m(E)<\infty \) and \(f:E\to [0,M]\) is measurable, nonnegative, and bounded, then the above definition coincides with that in Section 4.2.
[Chapter 4, Problem 24] Let \(E\subseteq \R \) be measurable and let \(f:E\to [0,\infty ]\) be measurable. Then
Chebychev’s inequality. Let \(f\) be a nonnegative measurable function on a measurable set \(E\). Let \(\lambda >0\). Then
(Problem 1260) Prove Chebychev’s Inequality.
Proposition 4.9. Let \(f\) be a nonnegative measurable function on a measurable set \(E\). Then \(\int _E f=0\) if and only if \(f(x)=0\) for almost every \(x\in E\).
(Problem 1270) Prove Proposition 4.9.
Theorem 4.10. Let \(f\) and \(g\) be nonnegative measurable functions defined on a measurable set \(E\).
(Problem 1280) Prove Theorem 4.10, part (i).
(Problem 1281) Prove Theorem 4.10, part (ii).
Theorem 4.11. If \(A\) and \(B\) are two disjoint measurable sets and \(f:A\cup B\to \R \) is measurable and nonnegative, then \(\int _{A\cup B} f=\int _A f+\int _B f\). In particular, if \(m(E_0)=0\) and \(E_0\subseteq E\) for a measurable set \(E\), then \(\int _E f=\int _{E\setminus E_0} f\) for every nonnegative measurable function \(f:E\to [0,\infty ]\).
(Problem 1290) Prove Theorem 4.11.
Fatou’s lemma. Let \(E\subseteq \R \) be measurable and let \(\{f_n\}_{n=1}^\infty \) be a sequence of nonnegative measurable functions \(f_n:E\to [0,\infty ]\). Then
(Problem 1300) Prove Fatou’s lemma.
The monotone convergence theorem. Let \(E\subseteq \R \) be measurable and let \(\{f_n\}_{n=1}^\infty \) be a sequence of nonnegative measurable functions \(f_n:E\to [0,\infty ]\). Suppose in addition that \(f_n(x)\leq f_{n+1}(x)\) for all \(x\in E\). Then
(Problem 1310) Prove the monotone convergence theorem.
Corollary 4.12. Let \(E\subseteq \R \) be measurable and let \(\{f_n\}_{n=1}^\infty \) be a sequence of nonnegative measurable functions \(f_n:E\to [0,\infty ]\). Then
(Problem 1311) Prove Corollary 4.12.
[Definition: Integrable function] A nonnegative measurable function \(f\) on a measurable set \(E\) is said to be integrable, integrable over \(E\), or in \(L^1(E)\), if
Proposition 4.14. Let \(E\subseteq \R \) be measurable and let \(f:E\to [-\infty ,\infty ]\) be measurable. Then \(|f|\) is integrable (that is, \(\int _E |f|<\infty \)) if and only if both \(f^+\) and \(f^-\) are integrable.
[Definition: General Lebesgue integral] Suppose that \(f:E\to [-\infty ,\infty ]\) is measurable and that \(|f|\) is integrable. Then we say that \(f\) is integrable and that
(Problem 1312) Show that if \(f\) is integrable over \(E\) and nonnegative, then the above definition of \(\int _E f\) coincides with that in Section 4.3.
Proposition 4.13. If \(f\) is integrable over \(E\), then \(f\) is finite almost everywhere on \(E\).
(Problem 1320) Prove Proposition 4.13.
[Chapter 4, Problem 28] Let \(f\) be integrable over \(E\) and let \(C\) be a measurable subset of \(E\). Show that \(\int _C f =\int _E f\,\chi _C\).
Proposition 4.15. If \(f\) is integrable over \(E\), then \(\int _E f=\int _{E\setminus E_0} f\) whenever \(m(E_0)=0\).
(Problem 1321) Prove Proposition 4.15.
Proposition 4.16. (The integral comparison test.) Suppose that \(g\) is nonnegative and integrable over \(E\) and that \(|f|\leq g\) on \(E\). If \(f\) is measurable, then \(f\) is also integrable and \(\left |\int _E f\right |\leq \int _E |f|\leq \int _E g\).
(Problem 1330) Prove Proposition 4.16.
Theorem 4.17. Let \(f\) and \(g\) be functions integrable over a measurable set \(E\).
(Problem 1340) Prove Theorem 4.17, part (i).
(Problem 1350) Prove Theorem 4.17, part (ii).
Corollary 4.18. If \(A\) and \(B\) are two disjoint measurable sets and \(f:A\cup B\to \R \) is integrable over \(A\cup B\), then \(\int _{A\cup B} f=\int _A f+\int _B f\).
(Problem 1360) Prove Corollary 4.18.
The Lebesgue dominated convergence theorem. Let \(E\subseteq \R \) be measurable and let \(f\), \(f_n\), and \(g\) be measurable functions with domain \(E\). Suppose that \(g\) is nonnegative and integrable, that \(|f_n(x)|\leq g(x)\) for all \(n\in \N \) and almost every \(x\in E\), and that \(f_n\to f\) pointwise almost everywhere on \(E\). Then \(\int _E f_n\to \int _E f\).
(Problem 1370) Prove the Lebesgue dominated convergence theorem.
Theorem 4.20. Let \(\{E_n\}_{n=1}^\infty \) be a countable sequence of pairwise disjoint measurable sets. Let \(E=\bigcup _{n=1}^\infty E_n\). If \(f:E\to [-\infty ,\infty ]\) is integrable (that is, measurable and \(\int _E |f|<\infty \)), then
(Problem 1380) Prove Theorem 4.20.
Theorem 4.21. Let \(\{E_n\}_{n=1}^\infty \) be a countable sequence of measurable sets, let \(E=\bigcup _{n=1}^\infty E_n\), and suppose that \(f:E\to [-\infty ,\infty ]\) is integrable (that is, measurable and \(\int _E |f|<\infty \)).
Suppose that either:
\(E_n\subseteq E_{n+1}\) for all \(n\) and \(D=E=\bigcup _{n=1}^\infty E_n\).
\(E_n\supseteq E_{n+1}\) for all \(n\) and \(D=\bigcap _{n=1}^\infty E_n\).
Then
[Chapter 4, Problem 39] Prove Theorem 4.21.
Lemma 4.22. Let \(E\subset \R \) be measurable and suppose \(m(E)<\infty \). Let \(\delta >0\). Then there is a \(n\in \N \) and a list of pairwise disjoint sets \(E_1\), \(E_2,\dots ,E_n\) such that \(m(E_k)<\delta \) for all \(k\) and such that \(E=\bigcup _{k=1}^n E_k\).
(Problem 1390) Prove Lemma 4.22.
Proposition 4.23. Let \(E\subseteq \R \) be measurable and let \(f\) be a measurable function on \(E\).
(Problem 1400) Prove Proposition 4.23, part (a).
(Problem 1410) Prove Proposition 4.23, part (b).
[Definition: Uniformly integrable] Let \(E\subseteq \R \) be measurable and let \(\mathcal {F}\) be a family of measurable functions on \(E\). We say that \(\mathcal {F}\) is uniformly integrable over \(E\) if, for each \(\varepsilon >0\), there is a \(\delta >0\) such that, for all \(f\in \mathcal {F}\), we have that if \(A\subseteq E\) is measurable and \(m(A)<\delta \), then \(\int _A |f|<\varepsilon \).
(Problem 1411) Let \(E\subseteq \R \) be measurable and assume that \(m(E)<\infty \). Let \(f:E\to [-\infty ,\infty ]\). Then \(f\) is integrable over \(E\) if and only if \(\{f\}\) is uniformly integrable over \(E\).
(Problem 1420) Let \(E\subseteq \R \) be measurable and let \(g\) be integrable over \(E\). Show that \(\mathcal {F}=\{f|f:E\to [-\infty ,\infty ]\) is measurable and \(|f(x)|\leq |g(x)|\) for all \(x\in E\}\) is a uniformly integrable family.
Proposition 4.24. Any finite collection of integrable functions over a common domain \(E\) is uniformly integrable.
(Problem 1430) Let \(E\subseteq \R \) be measurable and let \(\mathcal {F}_1\), \(\mathcal {F}_2,\dots \mathcal {F}_n\) be a finite collection of families, each of which is uniformly integrable over \(E\). Show that \(\mathcal {F}=\bigcup _{k=1}^n \mathcal {F}_k\) is uniformly integrable over \(E\).
(Problem 1431) Prove Proposition 4.24.
Proposition 4.25. Let \(E\subset \R \) be measurable and assume \(m(E)<\infty \). Let \(\{f_n\}_{n=1}^\infty \) be a sequence of measurable functions on \(E\) and suppose that \(\mathcal {F}=\{f_n:n\in \N \}\) is uniformly integrable. Suppose that \(f_n\to f\) pointwise almost everywhere on \(E\) for some \(f\). Then \(f\) is integrable.
(Problem 1440) Prove Proposition 4.25.
(Problem 1460) Let \(E\), \(f_n\), and \(f\) be as in Proposition 4.25. Show that the family \(\{f\}\cup \{f_n:n\in \N \}\) is also uniformly integrable.
The Vitali convergence theorem. Let \(E\subset \R \) be measurable and assume \(m(E)<\infty \). Let \(\{f_n\}_{n=1}^\infty \) be a sequence of measurable functions on \(E\) and suppose that \(\mathcal {F}=\{f_n:n\in \N \}\) is uniformly integrable. Suppose that \(f_n\to f\) pointwise almost everywhere on \(E\) for some \(f\). Then \(\int _E f_n\to \int _E f\).
(Problem 1470) Prove the Vitali convergence theorem.
Theorem 4.26. Let \(E\subset \R \) be measurable and assume \(m(E)<\infty \). Suppose that \(\{h_n\}_{n=1}^\infty \) is a sequence of nonnegative integrable functions on \(E\) that converges pointwise almost everywhere to zero. Then \(\lim _{n\to \infty } \int _E h_n=0\) if and only if \(\{h_n:n\in \N \}\) is uniformly integrable over \(E\).
(Problem 1480) Prove Theorem 4.26.
Proposition 5.1. Let \(E\subseteq \R \) be measurable and let \(f\) be integrable over \(E\). Then for every \(\varepsilon >0\), there is an \(E_0\subseteq E\) with \(m(E_0)<\infty \) such that \(\int _{E\setminus E_0} |f|<\varepsilon \).
(Problem 1490) Prove Proposition 5.1.
[Definition: Tight] Let \(E\subseteq \R \) be measurable and let \(\mathcal {F}\) be a family of measurable functions on \(E\). We say that \(\mathcal {F}\) is tight if for each \(\varepsilon >0\), there is an \(E_0\subseteq E\) with \(m(E_0)<\infty \) and such that
(Problem 1491) Show that if \(m(E)<\infty \), then every family of measurable functions on \(E\) is tight.
(Problem 1500) By Proposition 4.23b, if \(m(E)<\infty \) and \(\mathcal {F}\) is a family of uniformly integrable functions over \(E\), then each \(f\in \mathcal {F}\) is in fact integrable. Give an example to show that this is not true if \(m(E)=\infty \) and then prove that if \(\mathcal {F}\) is both uniformly integrable and tight, then every element of \(\mathcal {F}\) is integrable.
The Vitali convergence theorem (infinite measure case). Let \(E\subseteq \R \) be measurable and let \(\mathcal {F}\) be a family of functions that is uniformly integrable and tight over \(E\). Suppose \(f_n\to f\) pointwise almost everywhere on \(E\). Then \(f\) is integrable over \(E\) and \(\lim _{n\to \infty }\int _E f_n=\int _E f\).
(Problem 1510) Prove the Vitali convergence theorem (infinite measure case).
Corollary 5.2. Let \(E\subseteq \R \) be measurable. Suppose that \(\{h_n\}_{n=1}^\infty \) is a sequence of nonnegative integrable functions on \(E\) that converges pointwise almost everywhere to zero. Then \(\lim _{n\to \infty } \int _E h_n=0\) if and only if \(\{h_n:n\in \N \}\) is uniformly integrable and tight over \(E\).
[Chapter 5, Problem 2] Prove Corollary 5.2.
[Definition: Convergence in measure. Let \(E\subseteq \R \) be measurable and let \(f_n\), \(f\) be measurable functions defined on \(E\)] Assume that \(f_n\) and \(f\) are finite almost everywhere. We say that the sequence \(\{f_n\}_{n=1}^\infty \) converges to \(f\) in measure if, for every \(\eta >0\), we have that
Proposition 5.3. Let \(E\subset \R \) be measurable and assume \(m(E)<\infty \). Let \(f_n\), \(f\) be measurable functions on \(E\) and assume that \(f_n\) and \(f\) are finite almost everywhere on \(E\). If \(f_n\to f\) pointwise almost everywhere on \(E\), then \(f_n\to f\) in measure on \(E\).
(Problem 1520) Prove Proposition 5.3.
(Problem 1530) If \(n\in \N \), then there is a unique \(k\in \N \cup \{0\}\) with \(2^k\leq n<2^{k+1}\). For each \(n\in N\), let \(I_n=[\frac {n}{2^k}-1,\frac {n+1}{2^k}-1]\). Let \(f_n=\chi _{I_n}\). Show that \(f_n\) converges to the zero function on \(E=[0,1]\) but that \(f_n(x)\not \to 0\) for any \(x\in [0,1]\).
(Problem 1540) Let \(\{f_n\}_{n=1}^\infty \) be as in the previous problem. Find a subsequence \(\{f_{n_k}\}_{k=1}^\infty \) that converges pointwise to zero everywhere.
Theorem 5.4. (Riesz) Let \(E\subseteq \R \) be measurable and let \(f_n\), \(f\) be measurable functions defined on \(E\). Suppose that \(f_n\to f\) in measure on \(E\). Show that there is a subsequence \(\{f_{n_k}\}_{k=1}^\infty \) that converges pointwise to \(f\) almost everywhere.
(Problem 1550) Prove Theorem 5.4.
Corollary 5.5. Let \(E\subset \R \) be measurable. Suppose that \(\{h_n\}_{n=1}^\infty \) is a sequence of nonnegative integrable functions on \(E\). Then \(\lim _{n\to \infty } \int _E h_n=0\) if and only if the following three conditions hold:
\(\{h_n:n\in \N \}\) is uniformly integrable over \(E\).
\(\{h_n:n\in \N \}\) is tight over \(E\).
\(h_n\to 0\) in measure on \(E\).
(Problem 1560) Begin the proof of Corollary 5.5 by assuming that \(\{h_n:n\in \N \}\) is uniformly integrable and tight and that \(h_n\to 0\) in measure, and showing that \(\int _E h_n\to 0\).
(Problem 1570) Complete the proof by showing that if \(\int _E h_n\to 0\) then \(h_n\to 0\) in measure on \(E\).
(Memory 1571) Let \((X,d)\) be a metric space and let \(f_1\), \(f_2,\dots ,f_n:X\to \R \) be continuous. Then \(f=\max \{f_1,f_2,\dots ,f_n\}\) is also continous on \(X\).
(Problem 1580) Let \((X,d)\) be a metric space and let \(\{g_n\}_{n=1}^\infty \) be a sequence of continuous functions \(g_n:X\to \R \). Let \(g:X\to (-\infty ,\infty ]\) be given by \(g(x)=\sup _{n\in \N } g_n(x)\). If \(g(x)<\infty \), show that \(g\) is lower semicontinuous at \(x\), that is, if \(\varepsilon >0\), then there is a \(\delta >0\) such that, if \(d(x,y)<\delta \), then \(g(y)>g(x)-\varepsilon \).
(Problem 1581) Let \((X,d)\) be a metric space and let \(\{h_n\}_{n=1}^\infty \) be a sequence of continuous functions \(h_n:X\to \R \). Let \(h:X\to (-\infty ,\infty ]\) be given by \(h(x)=\inf _{n\in \N } h_n(x)\). If \(h(x)>-\infty \), show that \(h\) is upper semicontinuous at \(x\), that is, if \(\varepsilon >0\), then there is a \(\delta >0\) such that, if \(d(x,y)<\delta \), then \(h(y)<h(x)+\varepsilon \).
Lemma 5.6. Let \(E\subseteq \R \) be measurable. For each \(n\in \N \), let \(\varphi _n:E\to [-\infty ,\infty ]\) and \(\psi _n:E\to [-\infty ,\infty ]\) be integrable.
Suppose that for each \(x\in E\) and each \(n\), \(k\in \N \), we have that \(\varphi _n(x)\leq \psi _k(x)\).
Suppose further that \(\lim _{n\to \infty } \int _E(\psi _n-\varphi _n)=0\).
Then \(\limsup _{n\to \infty } \varphi _n(x)=\liminf _{n\to \infty } \psi _n(x)\) for almost every \(x\in E\). Furthermore, if we define \(f(x)=\lim _{n\to \infty } \varphi _n(x)=\lim _{n\to \infty } \psi _n(x)\) for all such \(x\), then \(f\) is integrable and satisfies
(Problem 1590) Begin the proof of Lemma 5.6 by showing that \(\limsup _{n\to \infty } \varphi _n(x)=\liminf _{n\to \infty } \psi _n(x)\) for almost every \(x\in E\).
(Problem 1600) Complete the proof of Lemma 5.6 by showing that \(f\) is integrable and satisfies
Recall [Theorem 4.4]: . Let \(E\subset \R \) be measurable and suppose that \(m(E)<\infty \). Let \(f:E\to [-M,M]\) be a bounded function. If \(f\) is measurable, then \(f\) is Lebesgue integrable in the sense of Section 4.2, that is,
Theorem 5.7. Let \(E\subset \R \) be measurable and suppose that \(m(E)<\infty \). Let \(f:E\to [-M,M]\) be a bounded function.
Suppose that \(f\) is Lebesgue integrable in the sense of Section 4.2. Then \(f\) is measurable.
(Problem 1610) Prove Theorem 5.7.
Theorem 5.8. (Lebesgue) Let \([a,b]\subset \R \) be a closed and bounded interval and let \(f:[a,b]\to [-M,M]\) be a bounded function. Then \(f\) is Riemann integrable over \([a,b]\) if and only if
(Problem 1620) Give an example of a bounded function \(f:[0,1]\to \R \) that is not Riemann integrable, but such that there is a set \(E\subset [0,1]\) with \(m(E)=0\) and such that \(f\) is continuous on \([0,1]\setminus E\).
(The point of this problem is that the statement \(m(\{x\in [a,b]:f\text { is discontinuous at }x\})=0\) is much stronger than the statement that \(f\) is continuous on \([a,b]\setminus E\) for some \(E\) with \(m(E)=0\).)
(Problem 1630) In this problem we begin the proof of Theorem 5.8. Let \([a,b]\subset \R \) be a closed and bounded interval and let \(f:[a,b]\to [-M,M]\) be a bounded function. If \(n\in \N \), define \(a_{n,k}=a+k(b-a)2^{-n}\). Let \(\varphi _n\) and \(\psi _n\) be step functions which satisfy
Suppose that \(x\in (a,b)\setminus \{a_{n,k}:n,k\in \N ,0\leq k\leq 2^n\}\). Suppose further that \(f\) is continuous at \(x\). Show that \(\varphi _n(x)\to f(x)\).
(Problem 1631) Show that \(\psi _n(x)\to f(x)\).
(Problem 1640) Let \([a,b]\subset \R \) be a closed and bounded interval and let \(f:[a,b]\to [-M,M]\) be a bounded function. Suppose that \(f\) is continuous at \(x\) for almost every \(x\in [a,b]\). Show that \(f\) is Riemann integrable.
(Problem 1650) Let \([a,b]\subset \R \) be a closed and bounded interval and let \(f:[a,b]\to [-M,M]\) be a bounded function. Show that
(Problem 1660) Let \([a,b]\subset \R \) be a closed and bounded interval and let \(f:[a,b]\to [-M,M]\) be a bounded function. Show that
(Problem 1661) Let \([a,b]\subset \R \) be a closed and bounded interval and let \(f:[a,b]\to [-M,M]\) be a bounded function. Show that
(Problem 1662) Recall from Section 4.1 that \(f\) is Riemann integrable if
(Problem 1670) Let \([a,b]\subset \R \) be a closed and bounded interval and let \(f:[a,b]\to [-M,M]\) be a bounded function. Suppose that \(f\) is Riemann integrable. Show that \(f\) is continuous almost everywhere on \([a,b]\).
(Problem 1671) Let \(I\subseteq [-\infty ,\infty ]\) be an interval and let \(f:I\to [-\infty ,\infty ]\) be monotonic (either nonincreasing or nondecreasing).
If \(x\in I\), define \(f(x\pm )\) by
If \(x=\inf I\), then \(f(x-)=f(x)\).
If \(x=\sup I\), then \(f(x+)=f(x)\).
If \(x\in I\) with \(x>\inf I\), let \(f(x-)=\sup \{f(t):t\in I,\> t<x\}\) if \(f\) is nondecreasing and \(f(x-)=\inf \{f(t):t\in I,\> t<x\}\) if \(f\) is nonincreasing.
If \(x\in I\) with \(x<\sup I\), let \(f(x-)=\sup \{f(t):t\in I,\> t>x\}\) if \(f\) is nondecreasing and \(f(x-)=\inf \{f(t):t\in I,\> t>x\}\) if \(f\) is nonincreasing.
If \(x\), \(y\in I\), show that
\(f(x-)=\lim _{t\to x^-} f(t)\) if \(x>\inf I\),
\(f(x+)=\lim _{t\to x^+} f(t)\) if \(x<\sup I\), (and in particular these two limits exist),
If \(f(x-)<y<f(x+)\) then either \(f^{-1}(\{y\})=\emptyset \) or \(f^{-1}(\{y\})=\{x\}\).
If \(f\) is nondecreasing and \(x\lt y\), or if \(f\) is nonincreasing and \(f\gt y\), then \(f(x)\leq f(x+)\leq f(y-)\leq f(y)\).
\(f\) is continuous at \(x\) if and only if \(f(x-)=f(x+)\).
Theorem 6.1. Let \(a\), \(b\in [-\infty ,\infty ]\) with \(a<b\) and let \(f:(a,b)\to \R \) be monotonic. The set \(\{x\in (a,b):f\) is not continuous at \(x\}\) is at most countable.
(Problem 1680) Prove Theorem 6.1.
Proposition 6.2. Let \(a\), \(b\in \R \) with \(a<b\) and let \(C\subset (a,b)\). Then there exists a function \(f:(a,b)\to \R \) that is strictly increasing and such that \(C=\{x\in (a,b):f\) is not continuous at \(x\}\).
(Problem 1690) Prove Proposition 6.2.
[Definition: Open ball] If \((X,d)\) is a metric space, \(x\in X\), and \(r>0\), then the open ball centered at \(x\) of radius \(r\) is \( B(x,r)=\{y\in X:d(x,y)< r\}\).
[Definition: Closed ball] If \((X,d)\) is a metric space, \(x\in X\), and \(r>0\), then the closed ball centered at \(x\) of radius \(r\) is \(\overline B(x,r)=\{y\in X:d(x,y)\leq r\}\).
[Definition: Diameter] If \((X,d)\) is a metric space and \(Y\subseteq X\), then \(\diam Y=\sup \{d(x,y):x,y\in Y\}\).
(Problem 1691) If \((X,d)\) is a metric space, \(x\in X\), and \(r>0\), then \(r\geq \frac {1}{2}\diam \overline B(x,r)\).
[Definition: Separable] A metric space is separable if it contains a countable dense subset.
[Definition: Scaled ball] Let \((X,d)\) be a metric space. If \(x\in X\) and \(r\), \(s>0\), then \(sB(x,r)=B(x,r)\) and \(s\overline {B}(x,r)=\overline {B}(x,sr)\).
(Problem 1700) (The Vitali covering lemma, finite version.) Let \((X,d)\) be a metric space. Let \(\mathcal {F}\) be a finite collection of closed balls in \(X\). Let \(E\subseteq X\) satisfy \(E\subseteq \bigcup _{B\in \mathcal {F}} B\).
Show that there is a subset \(\mathcal {S}\subseteq \mathcal {F}\) such that
(Problem 1701) Let \((X,d)\) be a separable metric space, let \(B\) and \(\beta \) be two closed balls in \(X\), and suppose that \(B\cap \beta \neq \emptyset \) and there is a \(r>0\) such that \(\diam \beta \leq 2r\) and \( \diam B\geq r\). Show that \(\beta \subseteq 5B\).
(Problem 1710) Let \((X,d)\) be a separable metric space. Let \(\mathcal {C}\) be a (possibly infinite) collection of closed balls in \(X\). Suppose that there is an \(r>0\) such that \(2r\leq \diam B\leq 3r\) for all \(B\in \mathcal {C}\). Let \(E\subseteq X\) satisfy \(E\subseteq \bigcup _{B\in \mathcal {C}} B\).
Show that there is a countable subset \(\mathcal {S}\subseteq \mathcal {C}\) such that
(Bonus Problem 1711) Let \((X,d)\) be a metric space. Let \(\mathcal {C}\) be a (possibly infinite) collection of closed balls in \(X\). Suppose that there is an \(r>0\) such that \(r\leq \diam B\leq 2r\) for all \(B\in \mathcal {C}\). Let \(E\subseteq X\) satisfy \(E\subseteq \bigcup _{B\in \mathcal {C}} B\).
Use Zorn’s lemma to show that there is a subset \(\mathcal {S}\subseteq \mathcal {C}\) such that
(Problem 1720) Let \(\mathcal {S}\) be a collection of pairwise-disjoint closed balls (that is, closed bounded intervals) in \(\R \). Suppose that \(r=\inf \{\ell (I):I\in \mathcal {S}\}\) is positive. Suppose in addition that \(m^*(\bigcup _{I\in \mathcal {S}} I)<\infty \). Show that \(\mathcal {S}\) is finite.
(Problem 1721) Let \(\mathcal {S}\) be a (possibly infinite) collection of pairwise-disjoint closed balls (that is, closed bounded intervals) in \(\R \). Suppose that \(r=\inf \{\ell (I):I\in \mathcal {S}\}\) is positive. Show that \(\bigcup _{I\in \mathcal {S}} I\) is closed.
(Problem 1722) Let \((X,d)\) be a metric space. Let \(\mathcal {C}\) be a (possibly infinite) collection of closed balls in \(X\). Suppose that there is a \(R<\infty \) such that \(0<\diam (B)\leq R\) for all \(B\in \mathcal {C}\). Let \(E\subseteq X\) satisfy \(E\subseteq \bigcup _{B\in \mathcal {C}} B\).
For each \(k\geq 0\), let
Show that \(\mathcal {S}\subseteq \mathcal {C}\) and that, if \(B\), \(\beta \in \mathcal {S}\), then \(B\cap \beta =\emptyset \). (For partial credit, you may prove this under the assumption that \(X=\R \), \(d(x,y)=|x-y|\), and \(E\) is a measurable set of finite measure.)
(Problem 1730) (The Vitali covering lemma, infinite version.) Let \((X,d)\) be a metric space. Let \(\mathcal {C}\) be a (possibly infinite) collection of closed balls in \(X\). Suppose that \(\diam (B)>0\) for all \(B\in \mathcal {C}\) and that \(\sup _{B\in \mathcal {C}} \diam B<\infty \); however, we do not impose a lower bound on the diameters of the balls in \(\mathcal {C}\). Let \(E\subseteq X\) satisfy \(E\subseteq \bigcup _{B\in \mathcal {C}} B\).
Show that there is a subcollection \(\mathcal {S}\subseteq \mathcal {C}\) such that
(Problem 1740) (The Vitali covering lemma, small ball version.) Let \(E\subseteq \R \). Let \(\mathcal {C}\) be a (possibly infinite) collection of closed bounded intervals in \(\R \) such that \(\diam (I)>0\) for all \(I\in \mathcal {C}\) and such that, for all \(\delta >0\), we have that
The Vitali covering lemma, book version. Let \(E\subset \R \) satisfy \(m^*(E)<\infty \). Let \(\mathcal {C}\) be a (possibly infinite) collection of closed bounded intervals such that \(\diam (I)>0\) for all \(I\in \mathcal {C}\) and such that, for all \(\delta >0\), we have that
(Problem 1750) Prove the Vitali covering lemma, book version.
[Definition: The derivative] Let \(E\subseteq \R \) and let \(f:E\to \R \). Suppose that \(x\in E\) is an interior point, that is, \((x-\delta ,x+\delta )\subset E\) for some \(\delta >0\). We define
If \(\overline {D} f(x)=\underline {D} f(x)\) and is finite, we say that \(f\) is differentiable at \(x\) and define \(f'(x)=\overline {D} f(x)=\underline {D} f(x)\).
(Memory 1751) (The Mean Value Theorem). Suppose that \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\). Then there is an \(x\in (a,b)\) such that \(f(b)-f(a)=(b-a)f'(x)\). In particular, if \(f'(x)\geq \lambda \) for all \(x\in (a,b)\), then \(f(b)-f(a)\geq (b-a)\lambda \).
Lemma 6.3. Let \(f:[a,b]\to \R \) be nondecreasing for some \(a<b\), \(a\), \(b\in \R \). Let \(\lambda >0\). Then
(Problem 1760) Prove Lemma 6.3.
Lebesgue’s theorem. If \(f:(a,b)\to \R \) is monotonic, then \(f\) is differentiable almost everywhere in \((a,b)\).
(Problem 1770) Prove Lebesgue’s theorem.
[Definition: Divided difference and averaged function] Suppose \(f\) is integrable over \([a,b]\). Define
(Problem 1780) Show that \(\int _u^v \mathop {\mathrm {Diff}}\nolimits _h f = \mathop {\mathrm {Av}}\nolimits _h f(v)-\mathop {\mathrm {Av}}\nolimits _h f(u)\).
Corollary 6.4. If \(f\) is a nondecreasing real-valued function over \([a,b]\), then \(f'\) (which exists almost everywhere by Lebesgue’s theorem) is integrable over \([a,b]\) and
(Problem 1790) Prove Corollary 6.4.
(Problem 1800) Give an example of a function \(f\) that is an increasing function over \([a,b]\) and such that
(Problem 1801) Give an example of a continuous function \(f\) that is an increasing function over \([a,b]\) and such that
[Definition: Total variation] Let \([a,b]\subset \R \) be a closed bounded interval and let \(f:[a,b]\to \R \). The total variation of \(f\) over \([a,b]\) is
(Problem 1802) Show that we obtain an equivalent definition if we additionally require that \(a=x_0\) and \(b=x_n\).
(Problem 1810) If \(f:[a,b]\to \R \) is nondecreasing, show that \(TV(f)=f(b)-f(a)\).
(Problem 1820) If \(f:[a,b]\to \R \) is Lipschitz with Lipschitz constant \(M\), show that \(TV(f)\leq M(b-a)\).
(Problem 1830) If \(f:[a,b]\to \R \), \(g:[a,b]\to \R \), show that \(TV(f+g)\leq TV(f)+TV(g)\).
(Problem 1840) Give an example of a bounded, continuous function \(f\) from \([0,1]\) to \(\R \) such that \(TV(f)=\infty \).
(Problem 1841) Suppose that \(f:[a,b]\to \R \), that \(TV(f)<\infty \), and that \(a<c<b\). Show that \(TV(f\big \vert _{[a,c]})+TV(f\big \vert _{[c,b]})\leq TV(f)\).
(Problem 1842) Suppose that \(f:[a,b]\to \R \), that \(TV(f)<\infty \), and that \(a<c<b\). Show that \(TV(f\big \vert _{[a,c]})+TV(f\big \vert _{[c,b]})= TV(f)\).
(Problem 1850) Suppose that \(f:[a,b]\to \R \), that \(TV(f)<\infty \), and that \(a\leq c<d\leq b\). Show that \(TV(f\big \vert _{[c,d]})\leq TV(f)\). In particular, the function \(F(x)=TV(f\big \vert _{[a,x]})\) is nondecreasing.
Lemma 6.5. Let \(f:[a,b]\to \R \) and suppose that \(TV(f)<\infty \). Then \(f=g-h\), where \(g\) and \(h\) are both nondecreasing.
(Problem 1860) Prove Lemma 6.5.
Jordan’s Theorem. If \(f:[a,b]\to \R \), then \(f\) is of bounded variation (that is, \(TV(f)<\infty \)) if and only if \(f=g+h\), where \(g\) is nondecreasing and \(h\) is nonincreasing.
Corollary 6.6. If \(f\) is of bounded variation on the closed bounded interval \([a,b]\), then \(f\) is differentiable almost everywhere on \([a,b]\), and \(f'\) is integrable over \([a,b]\).
[Definition: Absolutely continuous] Let \([a,b]\subset \R \) be a closed and bounded interval and let \(f:[a,b]\to \R \) be a function.
We say that \(f\) is absolutely continuous if, for every \(\varepsilon >0\), there is a \(\delta >0\) such that, if \(n\in \N \) is an integer, and if the two lists of numbers \(\{a_k\}_{k=1}^n\) and \(\{b_k\}_{k=1}^n\) satisfy
\(a\leq a_k\leq b_k\leq b\) for all \(1\leq k\leq n\),
\(b_{k-1}\leq a_{k}\) for all \(2\leq k\leq n\),
\(\sum _{k=1}^n |b_k-a_k|<\delta \),
then
(Problem 1861) Show that every absolutely continuous function is uniformly continuous.
(Problem 1862) Give an example of a continuous function on a compact (closed and bounded) interval that is not absolutely continuous.
[Chapter 6, Problem 38a] We may replace \(n\) by \(\infty \) in the above definition and recover an equivalent definition; that is, \(f\) is absolutely continuous over \([a,b]\) if and only if, for every \(\varepsilon >0\), there is a \(\delta >0\) such that, if \(\{a_k\}_{k=1}^\infty \) and \(\{b_k\}_{k=1}^\infty \) satisfy
\(a\leq a_k\leq b_k\leq b\) for all \(1\leq k<\infty \),
\((a_k,b_k)\cap (a_j,b_j)\) for all \(j\), \(k\in \N \) with \(j\neq k\),
\(\sum _{k=1}^\infty |b_k-a_k|<\delta \),
then
(Problem 1870) Let \([a,b]\subseteq \mathbb {R}\) be a closed and bounded interval and let \(f:[a,b]\to \mathbb {R}\). Suppose that \(f'\) is defined almost everywhere on \([a,b]\) and is integrable on \([a,b]\), and that \(f(x)=f(a)+\int _a^x f'\) for all \(x\in [a,b]\). Show that \(f\) is absolutely continuous.
[Chapter 6, Problem 38b] Suppose that \(f\) satisfies the following weaker version of absolute continuity: for every \(\varepsilon >0\), there is a \(\delta >0\) such that, if \(\{a_k\}_{k=1}^n\) and \(\{b_k\}_{k=1}^n\) satisfy
\(a\leq a_k\leq b_k\leq b\) for all \(1\leq k\leq n\),
\(b_{k-1} < a_{k}\) for all \(2\leq k\leq n\),
\(\sum _{k=1}^n |b_k-a_k|<\delta \),
then
(Problem 1880) Suppose that \(f\) is absolutely continuous. Determine whether this implies that \(f\) also satisfies the following stronger version of absolute continuity: for every \(\varepsilon >0\), there is a \(\delta >0\) such that, if \(\{a_k\}_{k=1}^\infty \) and \(\{b_k\}_{k=1}^\infty \) satisfy
\(a\leq a_k\leq b_k\leq b\) for all \(1\leq k\leq n\),
\(\sum _{k=1}^n |b_k-a_k|<\delta \),
then \(\sum _{k=1}^\infty |f(b_k)-f(a_k)|<\varepsilon \).
Proposition 6.7. A Lipschitz continuous function on a closed bounded interval is absolutely continuous.
(Problem 1890) Prove Proposition 6.7.
Theorem 6.8. Let \([a,b]\subseteq \mathbb {R}\) be a closed and bounded interval and let \(f:[a,b]\to \mathbb {R}\). If \(f\) is absolutely continous, then \(f\) is of bounded variation and \(f=g-h\), where \(g\) and \(h\) are nondecreasing functions both of which are absolutely continuous on \([a,b]\).
(Problem 1900) Begin the proof of Theorem 6.8 by showing that, if \(f\) is absolutely continuous on the closed bounded interval \([a,b]\), then \(f\) is of bounded variation. Hint: Use Problem 1842.
(Problem 1901) Give an example of a function of bounded variation that is not absolutely continuous.
(Problem 1910) By Jordan’s Theorem and the previous problem, if \(f\) is absolutely continuous on the closed bounded interval \([a,b]\), then \(f=g-h\) where \(g\) and \(h\) are both nondecreasing functions. In particular, we may require that \(h(x)=TV(f\big \vert _{[a,x]})\). Show that \(h\) is absolutely continuous in addition to being nondecreasing.
(Problem 1920) Complete the proof of Theorem 6.8 by showing that the sum of two absolutely continuous functions is absolutely continuous.
Theorem 6.9. Let \([a,b]\subseteq \mathbb {R}\) be a closed and bounded interval and let \(f:[a,b]\to \mathbb {R}\). Suppose that \(f\) is continuous. Then \(f\) is absolutely continuous if and only if \(\{\Diff _h f:0<h\leq 1\}\) is a uniformly integrable family, where \(\Diff _h\) is as in Problem 1780.
(Problem 1930) Begin the proof of Theorem 6.9 by showing that, if \(\{\Diff _h f:0<h\leq 1\}\) is a uniformly integrable family, then \(f\) is absolutely continuous.
(Problem 1940) Complete the proof of Theorem 6.9.
(Problem 1950) Suppose that \(f:[a,b]\to \R \) is integrable over \([a,b]\) and continuous at \(a\) and \(b\). Show that
Theorem 6.10. Let \([a,b]\subseteq \mathbb {R}\) be a closed and bounded interval and let \(f:[a,b]\to \mathbb {R}\). Suppose that \(f\) is absolutely continuous.
Then \(f\) is differentiable almost everywhere on \((a,b)\), its derivative \(f'\) is integrable over \([a,b]\), and
(Problem 1960) Prove Theorem 6.10.
(Problem 1970) Let \([a,b]\subseteq \mathbb {R}\) be a closed and bounded interval and let \(f:[a,b]\to \mathbb {R}\). Show that \(f\) is absolutely continuous over \([a,b]\) if and only if \(f\) is differentiable almost everywhere in \((a,b)\), \(f'\) is integrable over \([a,b]\), and
Theorem 6.11. Let \([a,b]\subseteq \mathbb {R}\) be a closed and bounded interval and let \(f:[a,b]\to \mathbb {R}\). Then \(f\) is absolutely continuous if and only if there is a function \(g\) that is integrable over \([a,b]\) such that
(Problem 1980) Prove Theorem 6.11.
Corollary 6.12. Let \([a,b]\subseteq \mathbb {R}\) be a closed and bounded interval and let \(f:[a,b]\to \mathbb {R}\) be monotonic. Then \(f\) is absolutely continuous on \([a,b]\) if and only if
(Problem 1990) Prove Corollary 6.12.
Lemma 6.13. Let \([a,b]\subseteq \mathbb {R}\) be a closed and bounded interval and let \(f:[a,b]\to \mathbb {R}\) be integrable over \([a,b]\). Then \(f=0\) almost everywhere in \([a,b]\) if and only if \(\int _x^y f=0\) for all \(x\), \(y\) such that \(a<x<y<b\).
(Problem 2000) Prove Lemma 6.13.
Theorem 6.14. Let \([a,b]\subseteq \mathbb {R}\) be a closed and bounded interval and let \(f:[a,b]\to \mathbb {R}\) be integrable over \([a,b]\). Define
(Problem 2010) Prove Theorem 6.14.
(Problem 2020) Let \(I\subseteq \R \) be an open interval and let \(\varphi :I\to \R \) be a continuous function. Suppose that \(\varphi \) is twice continuously differentiable. If \(0<a<1\) and \(b=1-a\), show that
[Definition: Convex function] If \(I\subseteq \R \) is an open interval and \(\varphi :I\to \R \), then \(\varphi \) is said to be convex if
Proposition 6.15. If \(\varphi \) is differentiable on the open interval \(I\subseteq \R \) and \(\varphi '\) is nondecreasing, then \(\varphi \) is convex.
(Problem 2030) Prove Proposition 6.15.
The Chordal Slope Lemma. Let \(\varphi \) be convex on the open interval \(I\). If \(x\), \(y\), \(z\in I\) with \(x<y<z\), then
(Problem 2040) Prove the Chordal Slope Lemma. Do not assume that \(\varphi \) is differentiable.
[Definition: One-sided derivative] If \(E\subseteq \R \), and if \([x,x+\delta )\subseteq E\) for some \(x\in \R \) and some \(\delta >0\), and if \(\varphi :E\to \R \), we define the right derivative of \(\varphi \) at \(x\), denoted \(\varphi '(x^+)\), to be equal to the following limit (provided that the limit exists):
Lemma 6.16. Let \(\varphi \) be convex on the open interval \(I\). Then \(\varphi '(x^+)\) and \(\varphi '(x^-)\) exist for all \(x\in I\). Moreover, if \(u\), \(v\in I\) with \(u<v\), then
(Problem 2050) Prove Lemma 6.16.
Corollary 6.17. Let \(\varphi \) be convex on the open interval \(I\), where \(I\subseteq \R \) is an open interval. If \(K\subset I\) is a closed bounded interval, then \(\varphi \) is Lipschitz on \(K\).
(Problem 2060) Prove Corollary 6.17.
Theorem 6.18. Suppose that \(\varphi \) is convex on the open interval \(I\). Then \(\varphi \) is differentiable at all but countably many points, and its derivative \(\varphi '\) is an increasing function.
(Problem 2070) Prove Theorem 6.18.
Jensen’s Inequality. Let \(\varphi :\R \to \R \) be convex. Let \(f:[0,1]\to \R \) be integrable. Suppose that \(\varphi \circ f\) is also integrable over \([0,1]\). Then
(Problem 2080) Prove Jensen’s Inequality.
The Fundamental Theorem of Calculus. Suppose that \([a,b]\subset \R \) is a closed bounded interval, that \(F:[a,b]\to \R \) is continuous, that \(F'\) exists everywhere on \([a,b]\), and that \(F'\) is Riemann integrable over \([a,b]\). Then \(F(b)-F(a)=\int _a^b F'\).
Recall [Theorem 6.10]: Let \([a,b]\subseteq \mathbb {R}\) be a closed and bounded interval and let \(f:[a,b]\to \mathbb {R}\). Suppose that \(f\) is absolutely continuous.
Then \(f\) is differentiable almost everywhere on \((a,b)\), its derivative \(f'\) is integrable over \([a,b]\), and
(Problem 2090) Give an example of a closed and bounded interval \([a,b]\) and a continuous function \(f:[a,b]\to \R \) such that \(f'\) exists almost everywhere on \([a,b]\) and is Lebesgue integrable on \([a,b]\), but such that
(Problem 2091) Let \(A\), \(B\subseteq [-\infty ,\infty ]\) be two nonempty sets of extended real numbers. Suppose that \(a\leq b\) for all \(a\in A\) and all \(b\in B\). Then \(\sup A\leq \inf B\).
[Definition: Metric space] A metric space is a set \(X\) together with a function (metric) \(d:X\times X\to [0,\infty )\) such that
\(d(x,x)=0\) for all \(x\in X\),
if \(d(x,y)=0\) then \(x=y\),
\(d(x,y)=d(y,x)\) for all \(x\), \(y\in X\),
if \(x\), \(y\), \(z\in X\), then \(d(x,y)\leq d(x,z)+d(y,z)\).
[Definition: Vector space over \(\R \)] A real vector space is a set \(V\) together with two binary operations \(+:V\times V\to V\) and \(\cdot :\R \times V\to V\) such that, if \(u\), \(v\), \(w\in V\) and \(r\), \(s\in \R \), then
\(u+v=v+u\),
\((u+v)+w=u+(v+w)\),
If \(u+v=w+v\) then \(u=w\),
\(r\cdot (s\cdot v)=(rs)\cdot v\),
\(r\cdot (u+v)=r\cdot u+r\cdot v\),
\((r+s)\cdot v=r\cdot v+s\cdot v\),
\(1v=v\),
It is easy to show that \(0v=0w\) and \(v+0v\) for all \(v\), \(w\in V\); we let \(0_V=0v\) for any (all) \(v\in V\).
[Definition: Normed vector space] A normed vector space is a real vector space together with a metric \(d\) such that
\(d(v,w)=d(v-w,0)\),
\(d(rv,rw)=|r|d(v,w)\).
(We often write \(\|v\|=d(v,0)\).) We call \(d(v,0)\) the norm of \(v\).
(Problem 2092) Suppose that \(V\) is a real vector space and that \(\|\cdot \|:V\to \R \) is a function. Then \(\|\cdot \|\) is a norm on \(V\) (that is, the metric \(d(v,w)=\|v-w\|\) satisfies the above definition) if and only if
\(\|v\|\geq 0\) for all \(v\in V\),
\(\|v\|=0\) if and only if \(v=0_V\),
\(\|rv\|=|r|\,\|v\|\) for all \(r\in \R \) and all \(v\in V\),
\(\|v+w\|\leq \|v\|+\|w\|\) for all \(v\), \(w\in V\).
[Definition: Measurable and finite almost everywhere] Let \(E\subseteq \R \) be measurable and let \(\mathcal {F}\) be the set of all measurable functions \(f:E\to [-\infty ,\infty ]\) such that \(m(f^{-1}(\{-\infty ,\infty \}))=0\) (that is, \(m(f^{-1}(\R ))=m\{x\in E:f(x)\in \R \}=m(E)\)).
[Definition: Equal almost everywhere] Let \(f\), \(g\in \mathcal {F}\). We say that \(f\cong g\), or \(f=g\) almost everywhere, if \(m\{x\in E:f(x)\neq g(x)\}=0\).
(Problem 2093) Show that \(\cong \) is an equivalence relation and that, if \(f_1\cong f_2\) and \(g_1\cong g_2\), then
(Problem 2100) Suppose that \(f\), \(g\in \mathcal {F}\) and \(0<p<\infty \).
[Definition: Lebesgue (semi, quasi)-norm] Let \(0<p<\infty \) and let \(E\subseteq \R \) be measurable. If \(f\in \mathcal {F}\), we define
(Problem 2110) Let \(f:E\to [-\infty ,\infty ]\). Show that
In particular, show that the first set contains its infimum.
[Definition: Essential supremum] Let \(E\subseteq \R \) be measurable and let \(f:E\to \R \) be measurable. We define
(Problem 2120) Let \(E\subset \R \) be measurable with \(m(E)<\infty \) and let \(f:E\to \R \) be bounded. Show that
(Problem 2130) Let \(E\subset \R \) be measurable with \(m(E)<\infty \) and let \(f:E\to \R \) be bounded. Show that
[Definition: \(L^\infty \)] If \(f\in \mathcal {F}\), we let
[Definition: Lebesgue space] Let \(\mathcal {F}/\cong \) be the set of equivalence classes of \(\mathcal {F}\) modulo the equivalence relation \(\cong \). If \(0<p\leq \infty \), we let \(L^p(E)\) be the set of all elements \(f\) of \(\mathcal {F}/\cong \) such that some representative \(\tilde f:E\to [-\infty ,\infty ]\) satisfies \(\|\tilde f\|_{L^p(E)}<\infty \). We let \(\|f\|_{L^p(E)}=\|\tilde f\|_{L^p(E)}\).
(Problem 2131) If \(f\in L^p(E)\), show that \(\|f\|_{L^p(E)}\) is well defined.
[Definition: Sequence space] If \(0<p\leq \infty \), we let \(\ell ^p\) be the space of all sequences of real numbers \(\{a_n\}_{n=1}^\infty \) such that the \(\ell ^p\) norm
(Problem 2132) Let \(0<p\leq \infty \) and let \(E\subseteq \R \) be measurable. Show that:
\(\|f\|_{L^p(E)}\geq 0\) for all \(f\in L^p(E)\),
\(\|f\|_{L^p(E)}=0\) if and only if \(f=0\) as an element of \(L^p(E)\),
\(\|rf\|_{L^p(E)}=|r|\,\|f\|_{L^p(E)}\) for all \(r\in \R \) and all \(f\in L^p(E)\).
(Problem 2133) Let \(0<p\leq \infty \). Show that:
\(\|\{a_n\}_{n=1}^\infty \|_{\ell ^p}\geq 0\) for all \(\{a_n\}_{n=1}^\infty \in \ell ^p\),
\(\|\{a_n\}_{n=1}^\infty \|_{\ell ^p}=0\) if and only if \(a_n=0\) for all \(n\in \N \),
\(\|\{ra_n\}_{n=1}^\infty \|_{\ell ^p}=|r|\,\|\{a_n\}_{n=1}^\infty \|_{\ell ^p}\) for all \(r\in \R \) and all \(\{a_n\}_{n=1}^\infty \in \ell ^p\).
(Problem 2134) Show that \(L^1(E)\) and \(L^\infty (E)\) are normed vector spaces.
(Problem 2135) Show that \(\ell ^1\) and \(\ell ^\infty \) are normed vector spaces.
[Definition: Conjugate] If \(p\in [1,\infty ]\), we define the conjugate \(q\) of \(p\) to be the unique number in \([1,\infty ]\) that satisfies
(Problem 2136) Show that \(q=\frac {p}{p-1}\).
(Problem 2137) Show that \(pq=p+q\).
Young’s Inequality. If \(1<p<\infty \), \(0\leq a<\infty \), and \(0\leq b<\infty \), then
(Problem 2140) Prove Young’s Inequality.
Hölder’s inequality. Let \(E\subseteq \R \) be measurable, \(1\leq p\leq \infty \), and \(q\) be the conjugate of \(p\). If \(f\in L^p(E)\), \(g\in L^q(E)\), then \(fg\in L^1(E)\) and
(Problem 2150) Prove Hölder’s inequality in the case \(p=1\) or \(p=\infty \).
(Problem 2160) Prove Hölder’s inequality if \(1<p<\infty \).
Theorem 7.1. If \(E\subseteq \R \) is measurable, \(1\leq p<\infty \), and \(f\in L^p(E)\), then either \(f=0\) or the function \(f^*\) given by
(Problem 2170) Prove Theorem 7.1.
Minkowski’s Inequality. Let \(E\subseteq \R \) be measurable and \(1\leq p\leq \infty \). If \(f\), \(g\in L^p(E)\), then \(f+g\in L^p(E)\) and
(Problem 2180) Prove Minkowski’s Inequality in the cases \(p=1\) and \(p=\infty \).
(Problem 2190) Prove Minkowski’s Inequality in the case \(1<p<\infty \).
Corollary 7.2. Let \(E\subseteq \R \) be measuable and let \(1<p<\infty \). Let \(\mathcal {F}\subset L^p(E)\) and suppose that there is a \(M\in \R \) such that \(\sup _{f\in \mathcal {F}}\|f\|_p\leq M\). Then \(\mathcal {F}\) is uniformly integrable.
(Problem 2200) Prove Corollary 7.2.
Corollary 7.3. Let \(E\subseteq \R \) be measuable with \(m(E)<\infty \) and let \(1\leq p_1<p_2\leq \infty \). Then \(L^{p_2}(E)\subseteq L^{p_1}(E)\). Furthermore, if \(f\in L^{p_2}(E)\), then
(Problem 2210) Prove Corollary 7.3.
(Problem 2220) Let \(1\leq p_1<p_2\leq \infty \). Show that if \(\{a_n\}_{n=1}^\infty \in \ell ^{p_1}\), then
[Definition: Convergent sequence] Let \((X,d)\) be a metric space. If \(x\in X\) and \(x_n\in X\) for each \(n\in \N \), we say that \(x_n\) converges to \(x\) if \(\lim _{n\to \infty } d(x,x_n)=0\). In particular, if \((V,\|\,\|)\) is a normed linear space, \(v\in V\), and \(v_n\in V\) for each \(n\in \N \), we say that \(v_n\) converges to \(v\) if \(\lim _{n\to \infty } \|v-v_n\|=0\). If such an \(x\) (or \(v\)) exists, we call \(\{x_n\}_{n=1}^\infty \) (or \(\{v_n\}_{n=1}^\infty \)) a convergent sequence.
[Definition: Cauchy sequence] Let \((X,d)\) be a metric space. If \(x_n\in X\) for each \(n\in \N \), we say that \(\{x_n\}_{n=1}^\infty \) is a Cauchy sequence if \(\lim _{n\to \infty } \sup _{k\geq n} d(x_k,x_n)=0\). In particular, if \((V,\|\,\|)\) is a normed linear space, and \(v_n\in V\) for each \(n\in \N \), we say that \(\{v_n\}_{n=1}^\infty \) is a Cauchy sequence if \(\lim _{n\to \infty } \sup _{k\geq n} \|v_k-v_n\|=0\).
Proposition 7.4. Every convergent sequence is Cauchy. Every Cauchy sequence with a convergent subsequence is convergent.
(Problem 2221) Give an example of a normed vector space in which not all Cauchy sequences are convergent.
(Problem 2230) Give an example of a sequence of functions \(\{f_n\}_{n=1}^\infty \) such that each \(f_n\) is in \(L^2(\R )\cap L^3(\R )\), such that \(\{f_n\}_{n=1}^\infty \) is convergent in the \(L^2(\R )\)-norm, but such that \(\{f_n\}_{n=1}^\infty \) is not convergent in the \(L^3(\R )\)-norm.
The Riesz-Fischer Theorem. Let \(E\subseteq \R \) be measurable and let \(1\leq p\leq \infty \). Then \(L^p(E)\) is complete in the sense that every Cauchy sequence in \(L^p(E)\) is convergent.
Moreover, if \(f_n\to f\) in \(L^p(E)\), then there is a subsequence \(\{f_{n_k}\}_{k=1}^\infty \) such that \(f_{n_k}\to f\) pointwise almost everywhere.
[Chapter 7, Problem 33] The Riesz-Fischer theorem is true if \(p=\infty \). (We will prove the Riesz-Fischer theorem in the case \(1\leq p<\infty \) in Problem 2260 below; the next few theorems are important steps in the proof.)
[Definition: Rapidly Cauchy] Let \((V,\|\,\|)\) be a normed linear space and let \(\{v_n\}_{n=1}^\infty \) be a sequence of elements of \(V\). We say that \(\{v_n\}_{n=1}^\infty \) is rapidly Cauchy if
Proposition 7.5. Every rapidly Cauchy sequence (in any normed vector space) is Cauchy. Every Cauchy sequence (in any normed vector space) has a rapidly Cauchy subsequence.
(Problem 2240) Prove Proposition 7.5.
Theorem 7.6. Let \(E\subseteq \R \) be measurable and let \(1\leq p<\infty \). Let \(\{f_n\}_{n=1}^\infty \) be a rapidly Cauchy sequence in \(L^p(E)\).
Then there exists a \(f\in L^p(E)\) such that \(f_n\to f\) in \(L^p(E)\) and such that \(f_n(x)\to f(x)\) in \(\R \) for almost every \(x\in E\).
(Problem 2250) Begin the proof of Theorem 7.6 by showing that, if \(1\leq p<\infty \) and \(\{f_n\}_{n=1}^\infty \) is a rapidly Cauchy sequence in \(L^p(E)\), then \(\{f_n(x)\}_{n=1}^\infty \) is a Cauchy sequence of real numbers for almost every \(x\in E\). Hint: Use the Borel-Cantelli Lemma.
(Problem 2260) Prove Theorem 7.6. Then use Theorem 7.6 to prove the Riesz-Fischer Theorem.
(Problem 2261) Let \(f_n(x)=\frac {1}{1+(x-n)^2}\). Then \(f_n\to 0\) pointwise everywhere but \(f_n\not \to 0\) in \(L^p(\R )\).
(Problem 2262) Let \(f_n(x)=\frac {n^{3/p}}{1+n^6(x-1/n)^2}\). Then \(f_n\to 0\) pointwise everywhere but \(f_n\not \to 0\) in \(L^p([0,1])\).
Theorem 7.7. Let \(E\subseteq \R \) be measurable and let \(1\leq p<\infty \). Suppose that \(\{f_n\}_{n=1}^\infty \) is a sequence in \(L^p(E)\) that converges pointwise almost everywhere on \(E\) to some function \(f:E\to \R \). Then \(f_n\to f\) in \(L^p(E)\) if and only if \(\|f_n\|_p\to \|f\|_p\) in \(\R \).
(Problem 2270) Prove Theorem 7.7.
(Problem 2280) Let \(E\subseteq \R \) be measurable and let \(1\leq p<\infty \). Let \(\varphi \) be a simple function defined on \(E\). Show that \(\varphi \in L^p(E)\) if and only if \(\varphi \) is finitely supported, that is, if \(m(\{x\in E:\varphi (x)\neq 0\})<\infty \).
[Definition: Dense] A subset \(Y\) of a metric space \((X,d)\) is dense if, for all \(x\in X\) and all \(r>0\), there is a \(y\in Y\) with \(d(x,y)<r\). In particular, a subset \(A\) of a normed vector space \((V,\|\,\|)\) is dense if, for every \(v\in V\) and every \(r>0\), there is an \(a\in A\) with \(\|v-a\|<r\).
[Chapter 7, Problem 36] A subset \(Y\) of a metric space \((X,d)\) is dense if and only if, whenever \(x\in X\), there is a sequence \(\{y_n\}_{n=1}^\infty \) of points in \(y\) with \(y_n\to x\).
[Chapter 7, Problem 37] Let \((X,d)\) be a metric space and let \(Z\subseteq Y\subseteq X\). If \(Y\) is dense in \((X,d)\), and if \(Z\) is dense in the subspace \((Y,d)\), then \(Z\) is also dense in \((X,d)\).
[Definition: Step function] We say that \(\varphi :\R \to \R \) is a step function if there are finitely many points \(x_0<x_1<\dots <x_n\) such that \(\varphi \) is constant on \((-\infty ,x_0)\), on \((x_n,\infty )\), and on each of the intervals \((x_{k-1},x_k)\) for all \(1\leq k\leq n\). We say that \(\varphi \) is a compactly supported step function if in addition \(\varphi =0\) on \((-\infty ,x_0)\) and on \((x_n,\infty )\).
The set of finitely supported simple functions is dense in \(L^1(\R )\).
The set of compactly supported step functions is dense in \(L^1(\R )\).
The set of compactly supported continuous functions is dense in \(L^1(\R )\).
Proposition 7.9. Let \(E\subseteq \R \) be measurable and let \(1\leq p\leq \infty \).
If \(1\leq p<\infty \), then the set of finitely supported simple functions is dense in \(L^p(E)\).
If \(p=\infty \), then the set of simple functions is dense in \(E\).
(Problem 2290) Prove Proposition 7.9.
Proposition 7.10. The set of compactly supported step functions is dense in \(L^p(\R )\) for \(1\leq p<\infty \).
(Problem 2300) Prove Proposition 7.10.
Theorem 7.12. The set of compactly supported continuous functions is dense in \(L^p(\R )\) for all \(1\leq p<\infty \) (and therefore is dense in \(L^p(E)\) for any \(1\leq p<\infty \) and for any \(E\subseteq \R \) measurable).
(Problem 2310) Prove Theorem 7.12.
Theorem 7.11. The set of compactly supported step functions that take rational values and have discontinuities at rational points is dense \(L^p(\R )\) for any \(1\leq p<\infty \). Therefore, \(L^p(E)\) is separable for \(1\leq p<\infty \) and for any \(E\subseteq \R \) measurable.
(Problem 2320) Prove Theorem 7.11.
(Problem 2330) Show that \(\ell ^\infty \) is not separable.
(Problem 2340) Let \(E\subseteq \R \) be measurable and suppose that \(m(E)>0\). Show that \(L^\infty (E)\) is not separable.
[Definition: Linear functional] If \(X\) is a real vector space, a linear functional on \(X\) is a function \(T:X\to \R \) that satisfies
(Problem 2341) Let \(X=\R ^n\). Show that, if \(v\in \R ^n\), then \(T_v(x)=\langle v,x\rangle \) is a linear functional on \(X\), where \(\langle \,,\,\rangle \) denotes the usual inner product.
(Problem 2350) Let \(X=\R ^n\). Let \(T:X\to \R \) be a linear functional. Show that there is a \(v\in \R ^n\) such that \(T(x)=\langle v,x\rangle \) for all \(x\in X\).
[Definition: Bounded linear functional; norm of a linear functional] Let \(X\) be a normed real vector space and let \(T:X\to \R \) be a linear functional. If
(Problem 2351) Let \(T\) be a linear functional. Show that
where in the last expression we have that the infimum of the empty set is infinity, that is, \(\{M\in \R :|T(x)|\leq M\|x\|\text { for all }x\in X\}\) is empty if and only if the four suprema are infinite.
(Problem 2352) Give an example of a real vector space and a linear functional that is not bounded.
(Problem 2360) Let \(X\) be a normed real vector space (and therefore a metric space). Let \(T:X\to \R \) be a linear functional. Show that \(T\) is continuous if and only if \(T\) is bounded.
Proposition 8.3. Let \(X\) be a normed real vector space and let \(S\), \(T:X\to \R \) be two bounded linear functionals. Suppose that \(\{x\in X:T(x)=S(x)\}\) is dense in \(X\). Then \(T(x)=S(x)\) for all \(x\in X\).
[Definition: Dual space] Let \(X\) be a normed real vector space. Then the dual space \(X^*\) is the set of all bounded linear functionals on \(X\).
[Definition: Linear combinations of linear functions] If \(r\in \R \) and \(S\), \(T\) are linear functionals on a common vector space \(X\), then \(rT\) and \(S+T\) are also linear functionals on \(X\) defined by
Proposition 8.1. Let \(X\) be a normed real vector space. Then \(X^*\) is also a normed real vector space (with the norm in \(X^*\) given by \(\|\,\|_*\)).
Proposition 8.2. Let \(E\subseteq \R \) be measurable, \(1\leq p\leq \infty \), \(1/p+1/q=1\), and \(g\in L^q(E)\) (with \(q=\infty \) if \(p=1\)). Define \(T_g\) by
(Problem 2370) Prove Proposition 8.2 in the case \(p>1\).
(Problem 2371) Prove Proposition 8.2 in the case \(p=1\).
Lemma 8.4. Let \(E\subseteq \R \) be measurable and let \(1 \leq p \leq \infty \). Suppose that \(g:E\to [-\infty ,\infty ]\) is measurable and that there is a \(M\in [0,\infty )\) such that if \(\varphi \) is finitely supported and simple, then \(\varphi g\) is integrable and
Then \(g\in L^q(E)\) and \(\|g\|_q\leq M\), where \(1/p+1/q=1\).
(Problem 2380) Prove Lemma 8.4 in the case \(p=1\).
(Problem 2390) Prove Lemma 8.4 in the case \(p>1\).
Theorem 8.5. Let \(I\subset \R \) be a closed bounded interval and let \(1\leq p<\infty \). Let \(T\in (L^p(I))^*\). Then there is a \(g\in L^q(I)\), \(1/p+1/q=1\), such that
(Problem 2400) In this problem we begin the proof of Theorem 8.5. Let \(I=[a,b]\), \(T\), and \(p\) be as in Theorem 8.5. Define
(Problem 2410) Prove Theorem 8.5.
(Problem 2411) Let \(1\leq p<\infty \). Let \(T\in (L^p(\R ))^*\). Show that there is a \(g\in L^q(\R )\), \(1/p+1/q=1\), such that
(Problem 2412) Let \(1\leq p<\infty \). Let \(E\subseteq \R \) be measurable and let \(T\in (L^p(E))^*\). Show that there is a \(g\in L^q(E)\), \(1/p+1/q=1\), such that
The Riesz representation theorem for the dual of \(L^p(E)\). Let \(E\subseteq \R \) be measurable, \(1\leq p<\infty \), and \(1/p+1/q=1\).
Then there is a canonical isomorphism between \((L^p(E))^*\) and \(L^q(E)\): if we define \(\mathcal {R}_g(f)=\int _E fg\), then for every \(g\in L^q(E)\) we have that \(\mathcal {R}_g\in (L^p(E))^*\) with \(\|\mathcal {R}_g\|_*=\|g\|_q\), and conversely if \(T\in (L^p(E))^*\) then there is a \(g\in L^q(E)\) such that \(T=\mathcal {R}_g\) and \(\|g\|_q=\|T\|_*\). Furthermore, if \(\mathcal {R}_g=\mathcal {R}_h\) (in the sense of functions from \(L^p(E)\) to \(\R \)) then \(g=h\) as elements of \(L^q(E)\), that is, \(g(x)=h(x)\) for almost every \(x\in E\).
(Problem 2420) Prove the Riesz representation theorem for the dual of \(L^p(E)\).
Recall [Bolzano-Weierstrauß theorem]: If \(\{x_n\}_{n=1}^\infty \) is a bounded sequence of points in \(\R \), then there is a subsequence \(\{x_{n_k}\}_{k=1}^\infty \) that converges.
(Problem 2421) Let \(\{a_n\}_{n=1}^\infty \) be a sequence in \(\R \) and let \(\{a_{n_k}\}_{k=1}^\infty \) be a convergent subsequence. Show that
(Problem 2422) Let \(\{a_n\}_{n=1}^\infty \) be a sequence in \(\R \). Show that there is a subsequence \(\{a_{n_k}\}_{k=1}^\infty \) that converges to \(\liminf _{n\to \infty } a_{n}\).
[Definition: Weak convergence] Let \(X\) be a normed linear space, let \(x\in X\), and let \(\{x_n\}_{n=1}^\infty \) be a sequence in \(X\). We say that the sequence \(\{x_n\}_{n=1}^\infty \) converges weakly to \(x\), or \(x_n\rightharpoonup x\) in \(X\), if
Proposition 8.6. Let \(E\subseteq \R \) be measurable, let \(1\leq p<\infty \), and let\(1/p+1/q=1\). Let \(f\), \(f_n\in L^p(E)\). Then \(f_n\rightharpoonup f\) in \(L^p(E)\) if and only if
(Problem 2430) Give an example of a bounded sequence in \(L^p(\R )\), \(1\leq p\leq \infty \), that does not have a convergent subsequence.
(Problem 2440) Give an example of a measurable set \(E\subseteq \R \), a \(p\in (1,\infty )\), and functions \(f_n\), \(f\in L^p(E)\) such that \(f_n\rightharpoonup f\) weakly in \(L^p(E)\) but such that \(f_n\not \to f\) strongly in \(L^p(E)\).
(Problem 2441) Show that if \(f_n\to f\) (meaning \(\{f_n\}_{n=1}^\infty \) converges strongly to \(f\)) then \(\|f_n\|\to \|f\|\). Give an example to show that the condition \(f_n\rightharpoonup f\) (meaning \(\{f_n\}_{n=1}^\infty \) converges weakly to \(f\)) does not imply that \(\|f_n\|\to \|f\|\).
[Homework 16.1] Let \(1\leq p\leq \infty \) and let \(q\) satisfy \(1/p+1/q=1\). Let \(E\subseteq \mathbb {R}\) be measurable.
Suppose that \(f\in L^p(E)\) and that \(\int _E fg=0\) for all \(g\in L^q(E)\). Show that \(f=0\) almost everywhere in \(E\).
(Problem 2442) Let \(E\subseteq \R \) be measurable, let \(1\leq p<\infty \), and let \(1/p+1/q=1\). Let \(f\), \(F\), \(f_n\in L^p(E)\). Suppose that \(f_n\rightharpoonup f\) and \(f_n\rightharpoonup F\). Show that \(f=F\) in \(L^p(E)\).
Theorem 8.7. Let \(E\subseteq \R \) be measurable and let \(1\leq p<\infty \). Suppose that \(f_n\rightharpoonup f\) in \(L^p(E)\) (that is, \(\{f_n\}_{n=1}^\infty \) converges weakly to \(f\) in \(L^p(E)\)).
Then \(\{f_n\}_{n=1}^\infty \) is bounded in \(L^p(E)\) (that is, \(\sup _{n\in \N } \|f_n\|_p<\infty \)) and \(\|f\|_p\leq \liminf _{n\to \infty } \|f_n\|_p\).
(Problem 2443) This problem is a useful first step in the proof of Theorem 8.7. Let \(E\subseteq \R \) be measurable and let \(1\leq p<\infty \). Let \(\{f_n\}_{n=1}^\infty \) be a sequence in \(L^p(E)\) and let \(\{\varepsilon _n\}_{n=1}^\infty \) be a sequence of positive real numbers. Show that there exists a sequence \(\{g_n\}_{n=1}^\infty \) in \(L^q(E)\) such that \(\|g_{n+1}-g_n\|_q\leq \varepsilon _{n+1}\) for each \(n\in \N \) and such that
[Chapter 8, Problem 18] Let \(\{a_n\}_{n=1}^\infty \) be a sequence of positive numbers with \(a_n\to \infty \). Let \(X\) be a normed linear space. Suppose that \(X\) contains an unbounded weakly convergent sequence; that is, suppose that there is an \(x\in X\) and a sequence \(\{x_n\}_{n=1}^\infty \) such that \(x_n\rightharpoonup x\) in \(X\) and \(\{\|x_n\|\}_{n=1}^\infty \) is unbounded.
Show that there exists a \(y\in X\) and a sequence \(\{y_n\}_{n=1}^\infty \) in \(X\) with \(\|y_n\|=a_n\) and with \(y_n\rightharpoonup y\). Hint: The easiest choice of \(y\) is \(y=0\).
(Problem 2450) Prove the bound \(\sup _{n\in \N } \|f_n\|_p<\infty \) in Theorem 8.7.
(Problem 2460) Prove the bound \(\|f\|_p\leq \liminf _{n\to \infty } \|f_n\|_p\) in Theorem 8.7.
Corollary 8.8. Let \(E\subseteq \R \) be measurable and let \(1\leq p<\infty \). Suppose that \(f_n\rightharpoonup f\) weakly in \(L^p(E)\). Let \(1/p+1/q=1\) and let \(g_n\to g\) strongly in \(L^q(E)\). Then
(Problem 2470) Prove Corollary 8.8.
(Problem 2480) Give an example of a measurable set \(E\subseteq \R \), a \(p\), \(q\in (1,\infty )\) with \(1/p+1/q=1\), and sequences \(\{f_n\}_{n=1}^\infty \) and \(\{g_n\}_{n=1}^\infty \) such that \(f_n\rightharpoonup f\) weakly in \(L^p(E)\) and \(g_n\rightharpoonup g\) weakly in \(L^q(E)\) but such that \(\int _E f_n\,g_n\) does not converge to \(\int _E f\,g\).
(Problem 2490) Let \(E\subseteq \R \) be measurable and let \(1\leq p<\infty \). Let \(1/p+1/q=1\) and let \(S\subset L^q(E)\) be dense. Suppose that \(f_n\), \(f\in L^p(E)\), that \(\sup _{n\in \N } \|f_n\|_p<\infty \), and that
Proposition 8.9. Let \(E\subseteq \R \) be measurable and let \(1\leq p<\infty \). Let \(1/p+1/q=1\) and let \(\mathcal {H}\subset L^q(E)\) be such that
Theorem 8.10. Let \(E\subseteq \R \) be measurable and let \(1\leq p<\infty \). Let \(f_n\), \(f\in L^p(E)\). Suppose that \(\sup _{n\in \N } \|f_n\|_p<\infty \) and that
Theorem 8.11. Let \(-\infty <a<b<\infty \) and let \(1< p<\infty \). Let \(f_n\), \(f\in L^p([a,b])\). Suppose that \(\sup _{n\in \N } \|f_n\|_p<\infty \) and that
(Problem 2500) Prove Proposition 8.9, Theorem 8.10, and Theorem 8.11.
The Riemann-Lebesgue lemma. If \(1\leq p<\infty \) and \(f_n(x)=\sin (nx)\), then \(f_n\rightharpoonup 0\) in \(L^p([-\pi ,\pi ])\).
(Problem 2510) Prove the Riemann-Lebesgue lemma.
(Problem 2520) Let \(1\leq p<\infty \) and let \(f_n(x)=n^{1/p}\,\chi _{[1/n,2/n]}\). Show that \(\|f_n\|_p=1\), that \(\{f_n\}_{n=1}^\infty \) converges to zero pointwise, that if \(1<p<\infty \) then \(\{f_n\}_{n=1}^\infty \) converges to zero weakly in \(L^p([0,1])\), but that if \(p=1\) then \(\{f_n\}_{n=1}^\infty \) does not converge to zero weakly in \(L^1([0,1])\).
Theorem 8.12. Let \(E\subseteq \R \) be measurable and let \(1<p<\infty \). Suppose that \(\{f_n\}_{n=1}^\infty \) is a bounded sequence in \(L^p(E)\) and that \(f_n\) converges pointwise almost everywhere in \(E\) to some function \(f\). Then \(f\in L^p(E)\) and \(\{f_n\}_{n=1}^\infty \) converges to \(f\) weakly in \(L^p(E)\).
(Problem 2530) Prove Theorem 8.12.
(Problem 2531) Give an example of a sequence of functions in \(L^p(E)\) that converge pointwise but not weakly.
The Radon-Riesz theorem. Let \(E\subseteq \R \) be measurable and let \(1<p<\infty \). Suppose that \(f_n\rightharpoonup f\) weakly in \(L^p(E)\). Then \(f_n\to f\) strongly in \(L^p(E)\) if and only if \(\lim _{n\to \infty }\|f_n\|_p= \|f\|_p\).
(Problem 2540) Prove the Radon-Riesz theorem in the case \(p=2\).
(Bonus Problem 2541) Prove the Radon-Riesz theorem in the cases \(1<p<2\) and/or \(2<p<\infty \).
(Problem 2542) Let \(\{a_n\}_{n=1}^\infty \) be a sequence in \(\R \). Show that there is a subsequence \(\{a_{n_k}\}_{k=1}^\infty \) that converges to \(\liminf _{n\to \infty } a_{n}\).
Corollary 8.13. Let \(E\subseteq \R \) be measurable and let \(1<p<\infty \). Suppose that \(f_n\rightharpoonup f\) weakly in \(L^p(E)\). Then \(\|f\|_p=\liminf _{n\to \infty } \|f_n\|_p\) if and only if a subsequence of \(\{f_n\}_{n=1}^\infty \) converges to \(f\) strongly in \(L^p(E)\).
(Problem 2550) Prove Corollary 8.13.
(Problem 2551) Let \(f_n(x)=1+\sin (nx)\). Show that \(\{f_n\}_{n=1}^\infty \) is a counterexample to the Radon-Riesz theorem in the case \(p=1\).
Helly’s Theorem. Let \(X\) be a separable normed linear space and let \(\{T_n\}_{n=1}^\infty \) be a sequence in the dual space \(X^*\). Suppose that \(\{T_n\}_{n=1}^\infty \) is bounded, that is, there is some \(M<\infty \) such that
Then there is a subsequence \(\{T_{n_k}\}_{k=1}^\infty \) of \(\{T_n\}_{n=1}^\infty \) and a \(T\in X^*\) such that
(Problem 2560) Prove Helly’s Theorem.
Theorem 8.14. Let \(E\subseteq \R \) be measurable and let \(1<p<\infty \). Let \(\{f_n\}_{n=1}^\infty \) be a bounded sequence in \(L^p(E)\). Then there is a subsequence \(\{f_{n_k}\}_{k=1}^\infty \) that converges weakly to some \(f\in L^p(E)\).
(Problem 2570) Prove Theorem 8.14.
(Problem 2580) Give an example that shows that Theorem 8.14 is not true for \(p=1\), that is, a bounded sequence in \(L^1(E)\) that has no weakly convergent subsequence.
[Definition: Weakly sequentially compact] A subset \(K\) of a normed linear space \(X\) is weakly sequentially compact in \(X\) if, for every sequence \(\{x_n\}_{n=1}^\infty \) in \(K\), there is a subsequence \(\{x_{n_k}\}_{k=1}^\infty \) that converges weakly to an element of \(K\).
Theorem 8.15. If \(E\subseteq \R \) is measurable and \(1<p<\infty \), then \(\{f\in L^p(E):\|f\|_p\leq 1\}\) is weakly sequentially compact in \(L^p(E)\).
(Problem 2581) Give an example of a measurable set \(E\subseteq \R \), a \(p\in (1,\infty )\), and a sequence \(\{f_n\}_{n=1}^\infty \) such that \(\{f_n\}_{n=1}^\infty \) converges weakly but such that no subsequence of \(\{f_n\}_{n=1}^\infty \) converges strongly.
The Banach-Saks theorem. Let \(E\subseteq \R \) be measurable and let \(1<p<\infty \). Let \(f_n\), \(f\in L^p(E)\) and suppose that \(f_n\rightharpoonup f\) weakly in \(L^p(E)\).
Then there is a subsequence \(\{f_{n_k}\}_{k=1}^\infty \) of \(\{f_n\}_{n=1}^\infty \) such that, if we define
(Problem 2590) Prove the Banach-Saks theorem in the case \(p=2\).
[Definition: Continuous function] If \((X,d)\) is a metric space, then a function \(T:X\to \R \) is continuous if, whenever \(\{x_n\}_{n=1}^\infty \) is a sequence in \(X\) that satisfies \(x_n\to x\) for some \(x\in X\), we have that \(T(x_n)\to T(x)\).
(Memory 2591) A linear functional on a normed linear space is continuous if and only if it is bounded.
[Definition: Closed set] If \((X,d)\) is a metric space, then a subset \(C\subseteq X\) is closed if, whenever \(\{x_n\}_{n=1}^\infty \) is a sequence in \(C\) that satisfies \(x_n\to x\) for some \(x\in X\), we have that \(x\in C\).
(Memory 2592) If \((K,d)\) is a compact metric space and \(f:K\to \R \) is continuous, then there is an \(x\in K\) such that \(f(x)\geq f(y)\) for all \(y\in K\).
The Heine-Borel theorem. If \(K\subset \R ^n\) is closed and bounded, then \(K\) is compact.
(Memory 2593) If \((K,d)\) is a compact metric space and \(\{x_n\}_{n=1}^\infty \) is a sequence in \(K\), then \(\{x_n\}_{n=1}^\infty \) has a convergent subsequence. (If \(K\subset \R ^n\) then this is the Bolzano-Weierstrauß theorem.)
(Problem 2600) Let \(\{a_n\}_{n=1}^\infty \) be a sequence in \(\R \). Define \(b_n\) by
(Problem 2610) Let \(E\subseteq \R \) be measurable and let \(1\leq p\leq \infty \). If we regard \(L^p(E)\) as a metric space with the metric \(d(f,g)=\|f-g\|_p\), show that
(Problem 2620) With \(E\) and \(p\) as before, show that \(\{f\in L^p(E):\|f\|_p\leq 1\}\) is not compact.
[Definition: Convex set] Let \(X\) be a linear space and let \(C\subseteq X\). We say that \(C\) is convex if, whenever \(f\), \(g\in C\), we have that \(\{\lambda f+(1-\lambda g):0\leq \lambda \leq 1\}\subseteq C\).
(Problem 2621) If \(r\in (0,\infty )\), show that \(\{x\in X:\|x\|\leq r\}\) is convex.
(Problem 2622) Show that the intersection of a collection of convex sets is convex.
Lemma 8.16a. Let \(E\subseteq \R \) be measurable and let \(1<p<\infty \). Suppose that \(C\subset L^p(E)\) is closed and convex. Suppose further that \(\{f_n\}_{n=1}^\infty \) is a sequence in \(C\) and that \(f_n\rightharpoonup f\) weakly in \(L^p(E)\) for some \(f\in L^p(E)\). Then in addition \(f\in C\).
(Problem 2630) Prove Lemma 8.16a.
(Problem 2631) Give an example to show that the assumption that \(C\) is closed in Lemma 8.16a is necessary.
(Problem 2640) Give an example to show that the assumption that \(C\) is convex in Lemma 8.16a is necessary.
[Definition: Convex functional] Let \(X\) be a linear space, let \(C\subseteq X\) be convex, and let \(T:C\to \R \). We do not assume that \(T\) is either linear or continuous. We say that \(T\) is convex if
(Problem 2641) Show that every linear functional is convex.
(Problem 2642) If \(X\) is a normed linear space, show that the functional \(T:X\to \R \) given by \(T(x)=\|x\|\) is convex but not linear.
Theorem 8.17. Let \(E\subseteq \R \) be measurable and let \(1<p<\infty \). Suppose that \(C\subset L^p(E)\) is closed, bounded, and convex. Suppose further that \(T:C\to \R \) is continuous and convex. Then there is a \(f\in C\) such that
Lemma 8.16b. Let \(E\subseteq \R \) be measurable and let \(1<p<\infty \). Suppose that \(C\subset L^p(E)\) is closed, bounded, and convex. Suppose further that \(T:C\to \R \) is continuous and convex. Suppose further that \(\{f_n\}_{n=1}^\infty \) is a sequence in \(C\) and that \(f_n\rightharpoonup f\) weakly in \(L^p(E)\) for some \(f\in L^p(E)\). Then \(f\in C\), and
(Problem 2650) Prove Lemma 8.16b. (Note that we will use Lemma 8.16b in the proof of Theorem 8.17, so you may not use Theorem 8.17 to prove Lemma 8.16b.)
By Problem 2542, there is a subsequence \(\{T(f_{n_\ell })\}_{\ell =1}^\infty \) of \(\{T(f_n)\}_{n=1}^\infty \) such that\begin{equation*}\lim _{\ell \to \infty } T(f_{n_\ell })=\liminf _{n\to \infty } T(f_n).\end{equation*}We furthermore have that \(f_{n_\ell }\rightharpoonup f\) weakly in \(L^p(E)\). Thus, there is a sequence of functions \(\{f_{n_\ell }\}_{\ell =1}^\infty \) in \(C\) that converges weakly to \(f\) and such that\begin{equation*}\lim _{\ell \to \infty } T(f_{n_\ell })=\liminf _{\ell \to \infty } T(f_{n_\ell })=\liminf _{n\to \infty } T(f_n).\end{equation*}It is inconvenient to work with subsequences, so at this point we will redefine \(f_\ell \) so that \(\{f_{\ell }\}_{\ell =1}^\infty \) now refers to the subsequence \(\{f_{n_\ell }\}_{\ell =1}^\infty \) of the original subsequence. This procedure happens fairly often at the beginning of complicated proofs and is traditionally described as follows:By passing to a subsequence (see Problem 2542), we may assume that
\begin{equation*}\lim _{n\to \infty } T(f_{n})=\liminf _{n\to \infty } T(f_n).\end{equation*}That \(f\in C\) is Lemma 8.16a. As in the proof of Lemma 8.16a, by the Banach-Saks theorem, there is a subsequence \(\{f_{\ell _k}\}_{k=1}^\infty \) of \(\{f_\ell \}_{\ell =1}^\infty \) such that if
\begin{equation*}\phi _k=\frac {1}{k}\sum _{j=1}^k f_{\ell _j}\end{equation*}then \(\phi _k\to f\) strongly in \(L^p(E)\). Because \(T\) is continuous \(T:C\to \R \) and \(\phi _k\), \(f\in C\), we have that\begin{equation*}T(f)=\lim _{k\to \infty } T(\phi _k).\end{equation*}A straightforward induction argument (similar to that in Lemma 8.16a) establishes that\begin{equation*}T(\phi _k)=T\Bigl (\frac {1}{k}\sum _{j=1}^k f_{\ell _j}\Bigr ) \leq \frac {1}{k}\sum _{j=1}^k T(f_{\ell _j}).\end{equation*}Because \(\{T(f_{\ell _j})\}_{j=1}^\infty \) is a subsequence of \(\{T(f_{\ell })\}_{\ell =1}^\infty \), and\begin{equation*}\lim _{\ell \to \infty } T(f_\ell )=\liminf _{\ell \to \infty } T(f_\ell ),\end{equation*}we may apply Problem 2600 to see that\begin{align*} T(f)&=\lim _{k\to \infty } T(\phi _k) \\&\leq \lim _{k\to \infty }\frac {1}{k}\sum _{j=1}^k T(f_{\ell _j}) = \lim _{k\to \infty } T(f_{\ell _k}) = \lim _{\ell \to \infty } T(f_{\ell }) \alignbreak = \liminf _{\ell \to \infty } T(f_{\ell }).\end{align*}(The second limit exists by the squeeze theorem.) This completes the proof.
(Problem 2660) Let \(T\) and \(C\) be as in Theorem 8.17. Prove that \(\{T(f):f\in C\}\) is bounded below.
(Problem 2670) Prove Theorem 8.17.
[Definition: Measure space] Recall the definition of a \(\sigma \)-algebra: let \(X\) be a set and let \(\M \subseteq 2^X\). We say that \(\M \) is a \(\sigma \)-algebra of subsets of \(X\), or a \(\sigma \)-algebra over \(X\), if
Recall the definition of a measure: if \(X\) is a set and \(\M \) is a \(\sigma \)-algebra of subsets of \(X\), then we call \((X,\M )\) a measurable space. A measure on a measurable space \((X,\M )\) is a function \(\mu \) such that:
A measure space is a triple \((X,\M ,\mu )\), where \(X\) is a set, \(\M \) is a \(\sigma \)-algebra over \(X\), and \(\mu \) is a measure on \((X,\M )\).
(Problem 2671) The following are measure spaces:
\((\R ,\mathcal {L},m)\), where \(\mathcal {L}\) is the set of Lebesgue measurable sets and where \(m\) denotes Lebesgue measure.
\((\R ,\mathcal {B},m)\), where \(\mathcal {B}\) is the set of Borel sets.
\((X,2^X,\eta )\), where \(X\) is any set and where \(\eta \) is given by \(\eta (S)=|S|\) if \(S\) is finite and \(\eta (S)=\infty \) if \(S\) is not finite.
\((X,\M ,\delta _{x_0})\), where \(X\) is any nonempty set, \(\M \) is any \(\sigma \)-algebra over \(X\), \(x_0\in X\), and \(\delta _{x_0}(S)=1\) if \(x_0\in S\) and \(\delta _{x_0}(S)=0\) if \(x_0\notin S\).
\((X,\mathcal {C},\mu )\), where \(X\) is any uncountable set,
[Chapter 17, Problem 6] Let \((X,\M ,\mu )\) be a measure space. If \(E\in \M \), then \((E,\M _E,\mu _E)\) is a measure space, where \(\M _E=\{S\cap E:S\in \M \}=\{S\in \M :S\subseteq E\}\) and \(\mu _E=\mu \big \vert _{\M _E}\).
(Problem 2672) Let \(f:\R \to [0,\infty ]\) be measurable. Define \(\mu (A)=\int _A f\) for all measurable sets \(A\subseteq \R \). Show that \(\mu \) is a measure on the \(\sigma \)-algebra \(\M \) of Lebesgue measurable subsets of \(\R \).
Proposition 17.1. Let \((X,\M ,\mu )\) be a measure space.
If \(\{E_k\}_{k=1}^n\subseteq \M \) is a finite collection of sets in \(\M \) and \(E_k\cap E_j=\emptyset \) for all \(j\neq k\), then
If \(A\), \(B\in \M \) with \(A\subseteq B\), then
If \(A\), \(B\in \M \) with \(A\subseteq B\) and \(\mu (A)<\infty \), then
If \(\{E_k\}_{k=1}^\infty \subseteq \M \), then
(Problem 2673) Prove Proposition 17.1.
Proposition 17.2. Let \((X,\M ,\mu )\) be a measure space.
(Problem 2674) Prove Proposition 17.2.
The Borel-Cantelli Lemma in general measure spaces. Let \((X,\M ,\mu )\) be a measure space. Let \(\{E_k\}_{k=1}^\infty \) be a sequence of sets in \(\M \). Suppose that \(\sum _{k=1}^\infty \mu (E_k)<\infty \). Then \(|\{k\in \N :x\in E_k\}|<\infty \) for almost every \(x\in X\).
(Problem 2675) Prove the Borel-Cantelli Lemma in general measure spaces.
[Definition: Finite measure] Let \((X,\M ,\mu )\) be a measure space. We say that \(\mu \) is finite if \(\mu (X)<\infty \). (Note that \(X\) need not be finite!)
[Definition: \(\sigma \)-finite measure]Let \((X,\M ,\mu )\) be a measure space. We say that \(\mu \) is finite if
(Problem 2680) Give an example of a measure space that is not \(\sigma \)-finite.
[Definition: Complete measure space] Let \((X,\M ,\mu )\) be a measure space. We say that \((X,\M ,\mu )\) is complete if, whenever \(E\in \M \), \(\mu (E)=0\), and \(D\subseteq E\), we have that \(D\in \M \).
(Problem 2690) Show that \((\R ,\mathcal {B},m)\) is not complete.
(Problem 2700) Let \((X,\M ,\mu )\) be a measure space that is not complete. How should we define the completion of \((X,\M ,\mu )\) and why should we define it that way?
[Definition: Outer measure] Let \(X\) be a set. An outer measure on \(2^X\) is a function \(\mu ^*\) such that
(Problem 2701) Show that if \(\mu \) is a measure on the measure space \((X,2^X)\), then \(\mu \) is an outer measure on \(2^X\).
(Problem 2702) Give an example of an outer measure that is not a measure.
[Chapter 17, Problem 20] If \(X\) is a nonempty set and \(\alpha \in [0,\infty ]\), then the function \(\eta :2^X\to [0,\infty ]\) given by \(\eta (\emptyset )=0\) and \(\eta (S)=\alpha \) for all nonempty subsets \(S\) of \(X\) is an outer measure. In particular, the constant function zero is an outer measure.
[Definition: Measurable] If \(\mu ^*\) is an outer measure on \(2^X\) and \(E\subseteq X\), we call \(E\) measurable with respect to \(\mu ^*\) if
(Problem 2703) A set \(E\subseteq X\) is measurable with respect to \(\mu ^*\) if and only if
Proposition 17.5. The union of a finite collection of measurable sets is measurable.
(Problem 2710) Prove Proposition 17.5.
Proposition 17.6. Let \(\mu ^*\) be an outer measure on \(2^X\), let \(A\subseteq X\), and let \(\{E_k\}_{k=1}^n\) be a finite collection such that each \(E_k\) is measurable with respect to \(\mu ^*\) and such that \(E_k\cap E_\ell =\emptyset \) if \(k\neq \ell \). Show that
(Problem 2720) Prove Proposition 17.6.
Proposition 17.7. The union of a countable collection of measurable sets is measurable.
(Problem 2721) Prove Proposition 17.6.
Theorem 17.8. Let \(\mu ^*\) be an outer measure on \(2^X\). Let \(\M =\{E\subseteq X:E\) is measurable with respect to \(\mu ^*\}\). Let \(\bar \mu =\mu ^*\big \vert _{\M }\).
Then \(\M \) is a \(\sigma \)-algebra over \(X\), and \((X,\M ,\bar \mu )\) is a complete measure space.
(Problem 2730) Let \(\mu ^*\) and \(\M \) be as in Theorem 17.8. Show that \(\M \) is a \(\sigma \)-algebra over \(X\) and that \((X,\M ,\bar \mu )\) is a measure space.
(Problem 2740) Complete the proof of Theorem 17.8 by showing that \((X,\M ,\bar \mu )\) is complete.
[Definition: Empty sum] If \(\S \) is a set and \(f:\S \to [-\infty ,\infty ]\) is a function, then \(\emptyset \) is a finite (therefore countable) subset of \(\S \) and
[Definition: Empty union] If \(\S \) is a collection of sets, then \(\emptyset \) is a finite (therefore countable) subset of \(\S \) and
[Definition: Supremum and infimum of the empty set] Recall that if \(A\subseteq [-\infty ,\infty ]\), then \(\sup A\) is the smallest element of \([-\infty ,\infty ]\) such that \(\sup A\geq x\) for all \(x\in A\), and \(\inf A\) is the largest element of \([-\infty ,\infty ]\) such that \(\inf A\leq x\) for all \(x\in A\). Applying these definitions with \(A=\emptyset \), we see that
[Definition: Addition and subtraction on the extended real numbers]
If \(r\), \(s\in \R \), then \(r+s\) and \(r-s\) take their usual values.
If \(r\in \R \), then \(\infty +r=\infty -r=\infty \).
If \(r\in \R \), then \(r-\infty =-\infty \).
\(\infty +\infty =\infty \).
\(-\infty -\infty =-\infty \).
\(\infty -\infty \) is undefined.
In particular, if \(r+s=t+s\), you can only conclude that \(r=t\) if \(s\) is finite.
[Definition: Outer measure induced by a set function] Let \(X\) be a set, let \(\mathcal {S}\subseteq 2^X\), and let \(\mu :\mathcal {S}\to [0,\infty ]\) be a function. Then the outer measure \(\mu ^*\) induced by \(\mu \) is defined by
Theorem 17.9. The outer measure induced by a set function is an outer measure.
(Problem 2750) Prove Theorem 17.9.
[Definition: Induced Carathéodory measure] If \(\mu ^*\) is the outer measure on \(2^X\) induced by the set function \(\mu \), and if \((X,\M ,\bar \mu )\) is defined from \(\mu ^*\) as in Theorem 17.8, we call \(\bar \mu \) the Carathéodory measure induced by \(\mu \).
(Problem 2751) The Lebesgue measure is the Carathéodory measure induced by \(\ell \), where \(\mathcal {I}\) is the collection of all bounded open intervals in \(\R \) and \(\ell :\mathcal {I}\to [0,\infty ]\) maps \(I\) to the length of \(I\).
[Definition: \(\mathcal {S}_{\sigma \delta }\)] Let \(\mathcal {S}\subseteq 2^X\) for some set \(X\). Then \(\mathcal {S}_\sigma \) is the collection of countable unions of elements of \(\mathcal {S}\) and \(\mathcal {S}_{\sigma \delta }\) is the collection of countable intersections of elements of \(\mathcal {S}_\sigma \).
(Problem 2760) Let \(X\) be a set and let \(\mathcal {S}\subseteq 2^X\). Let \(\mu :\mathcal {S}\to [0,\infty ]\) be a set function and let \(\mu ^*\) and \((X,\M ,\bar \mu )\) be the outer measure and Carathéodory measure space induced by \(\mu \).
If \(E\in X\) and \(\varepsilon >0\), show that there is an \(A\in \mathcal {S}_\sigma \) with \(E\subseteq A\) and with \(\mu ^*(A)<\mu ^*(E)+\varepsilon \).
Proposition 17.10. Let \(X\) be a set and let \(\mathcal {S}\subseteq 2^X\). Let \(\mu :\mathcal {S}\to [0,\infty ]\) be a set function and let \((X,\M ,\bar \mu )\) be the Carathéodory measure space induced by \(\mu \).
If \(E\subseteq X\) and \(\mu ^*(E)<\infty \), then there is an \(A\) with
(Problem 2770) Prove Proposition 17.10.
[Definition: Arc length] If \((X,d)\) is a metric space and \(\gamma :[0,1]\to X\) is a continuous function, we define its arc length by
(Memory 2771) If \(\gamma :[0,1]\to \R ^n\) is continuously differentiable with bounded derivative, then
(Problem 2780) Let \(X\) be a set. Let \(\mathcal {I}\) be an index set and let \(\mu ^{(i)}\) be an outer measure on \(2^X\) for each \(i\in \mathcal {I}\). Show that \(\mu ^*\) given by \(\mu ^*(S)=\sup _{i\in \mathcal {I}}\mu ^{(i)}(S)\) is also an outer measure.
[Definition: Hausdorff measure] Let \((X,d)\) be a metric space. Let \(\alpha \geq 0\). If \(\delta >0\), define
We let \(H_\alpha ^{(\delta )}\) be the outer measure induced by the set function \(h_\alpha ^{(\delta )}\) and let the Hausdorff outer measure of dimension \(\alpha \) on \((X,d)\) be given by \(H^*_\alpha =\sup _{\delta >0} H^{(\delta )}_\alpha \).
(Problem 2790) Show that \(H_\alpha ^{(\delta )}\) is decreasing in \(\delta \), that is, if \(0<\delta _1<\delta _2\) and \(Y\subseteq X\) then \(H_\alpha ^{(\delta _1)}(Y)\geq H_\alpha ^{(\delta _2)}(Y)\). Conclude that \(H^*_\alpha (Y)=\lim _{\delta \to 0^+} H^{(\delta )}_\alpha (Y)\).
Proposition 20.30. Let \(0<\alpha <\beta \) and let \(Y\subseteq X\) for some metric space \((X,d)\). If \(H^*_\alpha (Y)<\infty \), then \(H^*_\beta (Y)=0\).
(Problem 2800) Prove Proposition 20.30.
(Problem 2810) If \(X=\R \) and \(d(x,y)=|x-y|\), show that \(H_1^{(\delta )}(Y)=m^*(Y)\) for all \(Y\subseteq \R \) and all \(\delta >0\). Conclude that \(H_1^*=m^*\).
(Problem 2820) Define \(\mathcal {O}_\delta = \{E\subseteq X:E\) is open and \(0\leq \diam (E)<\delta \}\). Let \(\tilde H_\alpha ^{(\delta )}\) be the outer measure induced by the set function \(h_\alpha ^{(\delta )}\big \vert _{\mathcal {O}_\delta }\) (that is, by \(h_\alpha ^{(\delta )}\) restricted to \(\mathcal {O}_\delta \)). Show that \(\tilde H_\alpha ^{(\delta )}=H_\alpha ^{(\delta )}\).
[Chapter 20, Problem 54] If \(\gamma :[0,1]\to X\) is a continuous curve that is one-to-one, show that \(\ell (\gamma )=H_1^*(\gamma ([0,1]))\).
(Problem 2830) If \((X,d)\) is a metric space, show that \(H_0^*\) is the counting measure.
(Bonus Problem 2831) Let \(X=\R ^n\) and let
(Problem 2840) Show that this is not true for the Hausdorff measure \(H_\alpha ^{(\delta )}\); that is, give an example of a set \(Y\) of diameter less than \(\delta \) such that \(h_\alpha ^{(\delta )}(Y)=\diam (Y)^\alpha \) is not equal to \(H_\alpha ^{(\delta )}(Y)\).
[Chapter 17, Problem 27] Let \(\mathcal {S}\) be a collection of subsets of a set \(X\), and let \(\mu :\mathcal {S}\to [0,\infty ]\) be a set function. Let \(\mu ^*\) be the induced outer measure on \(2^X\). Show that \(\mu ^*(E)=\mu (E)\) for all \(E\in \mathcal {S}\) if and only if \(\mu \) is countably monotone, that is, if and only if
[Definition: Premeasure]Let \(X\) be a set and let \(\S \subseteq 2^X\). Let \(\mu :S\to [0,\infty ]\). We call \(\mu \) a premeasure if:
(Problem 2841) Let \(\S =\{\{1,2\},\{3,4\},\{2,3\}\}\) and define \(\mu :\S \to [0,\infty ]\) by
\(\mu (\{1,2\})=\mu (\{3,4\})=1\),
\(\mu (\{2,3\})=5\).
Show that \(\mu \) is a premeasure but that \(\mu (\{2,3\})\neq \mu ^*(\{2,3\})\).
Proposition 17.11. Let \(\mu :\S \to [0,\infty ]\) be a set function, let \(\mu ^*\) be the outer measure induced by \(\mu \), let \(\bar \mu \) be the measure induced by \(\mu ^*\) and let \(\M \) be the \(\sigma \)-algebra of sets measurable with respect to \(\mu ^*\). Suppose that \(\S \subseteq \M \) and that \(\mu (Y)=\bar \mu (Y)\) for all \(Y\in \S \). Then \(\mu \) is a premeasure.
(Problem 2850) Prove Proposition 17.11.
[Definition: Closed with respect to the formation of relative complements] Let \(X\) be a set and let \(\S \subseteq 2^X\). We say that \(\S \) is closed with respect to the formation of relative complements if, whenever \(A\), \(B\in \S \), we have that \(A\setminus B\in \S \).
[Definition: Closed under finite intersections] Let \(X\) be a set and let \(\S \subseteq 2^X\). We say that \(\S \) is closed under finite intersections if, whenever \(A\), \(B\in \S \), we have that \(A\cap B\in \S \).
(Problem 2860) Show that, if \(\S \) is closed with respect to the formation of relative complements, then \(\S \) is closed under finite intersection. Give an example to show that the reverse is not true.
Theorem 17.12. Let \(X\) be a set and let \(\S \subseteq 2^X\). Let \(\mu :\S \to [0,\infty ]\). Suppose that
\(\S \) is closed with respect to the formation of relative complements.
\(\mu \) is a premeasure.
Let \(\mu ^*\) be the outer measure induced by \(\mu \), let \(\bar \mu \) be the measure induced by \(\mu ^*\) and let \(\M \) be the \(\sigma \)-algebra of sets measurable with respect to \(\mu ^*\). Then \(\S \subseteq \M \) and \(\mu (Y)=\bar \mu (Y)\) for all \(Y\in \S \).
(Problem 2870) Begin the proof of Theorem 17.12 by showing that all elements of \(\S \) are measurable.
(Problem 2880) Complete the proof of Theorem 17.12.
(Problem 2881) Let \(\S =\{\{1,2\},\{3,4\},\{2,3\}\}\) and define \(\mu :\S \to [0,\infty ]\) by
\(\mu (\{1,2\})=\mu (\{3,4\})=1\),
\(\mu (\{2,3\})=5\).
Show that \(\mu \) is a premeasure but that \(\mu (\{2,3\})\neq \mu ^*(\{2,3\})\).
[Definition: Ring] Let \(X\) be a set and let \(\S \subseteq 2^X\) with \(\S \neq \emptyset \). We say that \(\S \) is a ring if
\(\S \) is closed under finite unions.
\(\S \) is closed under the formation of relative complements: if \(A\), \(B\in \S \), then \(A\setminus B\in \S \). (By Problem 2860, this implies that \(\S \) is closed under finite intersections.)
If \(X\in \S \) we say that \(\S \) is an algebra.
(Bonus Problem 2882) Let \(\S \) be a collection of sets. Equip \(\S \) with the operations \(\oplus \), \(\otimes \) defined by \(A\oplus B=(A\setminus B)\cup (B\setminus A)\) and \(A\otimes B=A\cap B\). Show that \(\S \) is a ring in the sense defined above if and only if \((\S ,\oplus ,\otimes )\) is a ring (possibly without multiplicative identity) in the sense of MATH 51203. (\(\S \) is an algebra in the sense given above if and only if \((\S ,\oplus ,\otimes )\) is a ring with multiplicative identity.)
[Definition: Semiring] Let \(X\) be a set and let \(\S \subseteq 2^X\) with \(\S \neq \emptyset \). We say that \(\S \) is a semiring if
\(\S \) is closed under finite intersections.
If \(A\), \(B\in \S \), then \(A\setminus B=\bigcup _{k=1}^n C_k\), where the \(C_k\)s are pairwise disjoint elements of \(\S \).
If \(X\in \S \) we say that \(\S \) is a semialgebra.
(Problem 2883) Give an example of a semiring that is not a ring.
Proposition 17.13a. Let \(\S \) be a semiring of subsets of \(X\). Let
(Problem 2890) Prove Proposition 17.13a.
Proposition 17.13b. Let \(\S \), \(\S '\), and \(X\) be as in Proposition 17.13a. Let \(\mu :\S \to [0,\infty ]\) be a premeasure. Then there is a unique \(\mu ':\S '\to [0,\infty ]\) that is also a premeasure and satisfies \(\mu '(E)=\mu (E)\) for all \(E\in \S \).
(Problem 2900) Prove Proposition 17.13b.
(Problem 2901) Please carefully review the definitions in the Undergraduate Analysis presection of Section 17.4.
The Carathéodory-Hahn Theorem. Let \(\S \) be a semiring of subsets of \(X\) and let \(\mu :\S \to [0,\infty ]\) be a premeasure. Then the Carathéodory measure \(\bar \mu \) induced by \(\mu \) is an extension of \(\mu \) in the sense that every \(E\in \S \) is measurable with respect to \(\mu ^*\) and \(\mu (E)=\bar \mu (E)\) for all \(E\in \S \).
Furthermore, if \(\mu \) is \(\sigma \)-finite, (where we use the obvious definition of a \(\sigma \)-finite premeasure), then \(\bar \mu \) is also \(\sigma \)-finite, and is also the only measure on the \(\sigma \)-algebra of \(\mu ^*\)-measurable sets that extends \(\mu \).
(Problem 2902) Begin the proof of the Carathéodory-Hahn Theorem by showing that every \(E\in \S \) is measurable with respect to \(\mu ^*\) and that \(\mu (E)=\bar \mu (E)\) for all \(E\in \S \).
(Problem 2903) Let \(\M \) be the collection of subsets of \(X\) that are measurable with respect to \(\mu ^*\). Suppose that \(\nu :\M \to [0,\infty ]\) is a measure that satisfies \(\nu (E)=\mu (E)\) for all \(E\in \S \). Show that \(\nu (E)=\bar \mu (E)\) for all \(E\in \S _\sigma \), where \(\S _\sigma \) is the collection of countable unions of elements of \(\mathcal {S}\).
(Problem 2904) Let \(Y\in \S _\sigma \) satisfy \(\bar \mu (Y)<\infty \). Suppose that \(E\in \S _{\sigma \delta }\) and that \(E\subseteq Y\). Show that \(\bar \mu (E)=\nu (E)\). (Here \(\S _{\sigma \delta }\) is the collection of countable intersections of elements of \(\mathcal {S}_\sigma \).)
(Problem 2905) Let \(Y\in \S _\sigma \) satisfy \(\bar \mu (Y)<\infty \). Suppose that \(E\in \M \) and that \(E\subseteq Y\). Show that \(\bar \mu (E)=\nu (E)\).
(Problem 2910) Complete the proof of the Carathéodory-Hahn Theorem.
Corollary 17.14. Let \(\S \) be a semiring of subsets of \(X\) and let \(\B \) be the smallest \(\sigma \)-algebra of subsets of \(X\) that contains \(\S \). If \(\nu \), \(\lambda :\B \to [0,\infty ]\) are \(\sigma \)-finite measures, then \(\nu =\lambda \) if and only if \(\nu \big \vert _S=\lambda \big \vert _\S \).
[Homework 21.2] Let \(X\) be a set and let \(f:X\to [0,\infty ]\). Define \(\mu :2^X\to [0,\infty ]\) by
(Problem 2911) Show that \(I\) is a bounded interval if and only if \(I\) satisfies one of the following six conditions
\(I=\emptyset \) is empty,
\(I\) is a singleton set \(I=\{r\}\) for some \(r\in \R \),
\(I=(a,b)=\{c\in \R :a<c<b\}\) for some \(a\), \(b\in \R \) with \(a<b\),
\(I=(a,b]=\{c\in \R :a<c\leq b\}\) for some \(a\), \(b\in \R \) with \(a<b\),
\(I=[a,b)=\{c\in \R :a\leq c<b\}\) for some \(a\), \(b\in \R \) with \(a<b\), or
\(I=[a,b]=\{c\in \R :a\leq c\leq b\}\) for some \(a\), \(b\in \R \) with \(a<b\).
Proposition 20.10. Define \(\vol _n:\S \to [0,\infty ]\) by
(Problem 2920) Prove Proposition 20.10.
[Definition: Lebesgue measure on \(\R ^n\)] The outer measure \(\mu ^*_n\) induced by the premeasure \(\vol \) on the semiring \(\S \) of bounded rectangles in \(\R ^n\) is called the Lebesgue outer measure on \(\R ^n\), and the induced Carathéodory measure \(\mu _n\) is called the Lebesgue measure on \(\R ^n\), or \(n\)-dimensional Lebesgue measure.
Theorem 20.11. The \(\sigma \)-algebra \(\mathcal {L}^n\) of Lebesgue measurable subsets of \(\R ^n\) contains the Borel sets in \(\R ^n\) (the smallest \(\sigma \)-algebra of subsets of \(\R ^n\) that contains all the open sets). The Lebesgue measure is \(\sigma \)-finite, complete, and \(\mu _n(R)=\vol (R)\) for all bounded rectangles \(R\).
(Problem 2930) Prove Theorem 20.11.
[Definition: Measurable function] Let \((X,\M ,\mu )\) be a measure space and let \(E\in \mathcal {M}\) (that is, let \(E\) be measurable). Let \(f:E\to [-\infty ,\infty ]\). Suppose that for every \(c\in \R \) the set
Corollary 20.12. If \(E\subseteq \R ^n\) is Lebesgue measurable and \(f:E\to \R \) is continuous, then \(f\) is a Lebesgue measurable function.
(Problem 2931) Let \(X\) be a set. Let \(\mu ^*:2^X\to [0,\infty ]\) be an outer measure on \(2^X\), and let \(\bar \mu \) and \(\M \) be the induced measure and \(\sigma \)-algebra of measurable sets given by Theorem 17.8.
If \(\varphi :X\to [-\infty ,\infty ]\), show that \(\varphi \) is \(\M \)-measurable if and only if, for every \(c\in \R \) and every \(A\), \(B\subseteq X\) that satisfy
[Homework 23.1] Give an example of an outer measure \(\mu ^*:2^{\mathbb {R}}\to [0,\infty )\) such that \(\mu ^*(\mathbb {R})\lt \infty \) but such that
Proposition 20.27. Let \(X\) be a set. Let \(\mu ^*:2^X\to [0,\infty ]\) be an outer measure on \(2^X\), and let \(\bar \mu \) and \(\M \) be the induced measure and \(\sigma \)-algebra of measurable sets given by Theorem 17.8.
Let \(\varphi :X\to [-\infty ,\infty ]\). Then \(\varphi \) is \(\M \)-measurable if and only if, for all \(A\), \(B\subseteq X\) that satisfy
(Problem 2940) Prove Proposition 20.27.
[Definition: Carathéodory outer measure] Let \((X,d)\) be a metric space. An outer measure \(\mu ^*:2^X\to [0,\infty ]\) is a Carathéodory outer measure if
Theorem 20.28. If \(\mu ^*\) is a Carathéodory outer measure on a metric space \((X,d)\), then every Borel subset of \(X\) is measurable with respect to \(\mu ^*\).
(Problem 2950) Prove Theorem 20.28.
Proposition 20.29. If \((X,d)\) is a metric space and \(\alpha \geq 0\), then the Hausdorff outer measure \(H_\alpha ^*\) on \(X\) is a Carathéodory outer measure. Thus every Borel subset of \(X\) is Hausdorff measurable.
(Problem 2960) Prove Proposition 20.29.
Theorem 20.13. Let \(E\subseteq \R ^n\) be Lebesgue measurable. Then
(Problem 2970) Prove that \(\mu _n(E)=\inf \{\mu _n(\mathcal {O}):E\subseteq \mathcal {O},\>\mathcal {O}\text { is open}\}\).
(Problem 2980) Prove that \(\mu _n(E)=\sup \{\mu _n(K):K\subseteq E,\>K\text { is compact}\}\) in the case where \(E\) is bounded.
Corollary 20.14. Let \(E\subseteq \R ^n\). The following statements are equivalent.
[Definition: Infinite sum] Let \(\{a_k\}_{k=1}^\infty \) be a sequence of extended real numbers. If \(\sum _{k=1}^n a_k\) exists for all \(n\) (that is, if it is not the case that \(a_\ell =\infty \) and \(a_m=-\infty \) for some \(\ell \) and \(m\)), and if \(\lim _{n\to \infty } \sum _{k=1}^n a_k\) exists (in the sense of either finite or infinite limits), then we say that the series \(\sum _{k=1}^\infty a_k\) converges and write
[Definition: Absolute convergence] Let \(\{a_k\}_{k=1}^\infty \) be a sequence of real numbers. We say that \(\sum _{k=1}^\infty a_k\) converges absolutely if \(\sum _{k=1}^\infty |a_k|\) converges to a finite real number.
(Problem 2981) Show that \(\sum _{k=1}^\infty a_k\) converges absolutely if and only if \(\sup _{n\in \N } \sum _{k=1}^n | a_k|\) is finite.
[Definition: Unconditional convergence] Let \(\{a_n\}_{n=1}^\infty \) be a sequence of extended real numbers. We say that \(\sum _{n=1}^\infty a_n\) converges unconditionally if, for every bijection \(\sigma :\N \to \N \), we have that \(\sum _{n=1}^\infty a_{\sigma (n)}\) is convergent and
The Riemann rearrangement theorem. All absolutely convergent series converge unconditionally. All unconditionally convergent series of real numbers that converge to a finite value converge absolutely.
(Problem 2982) Suppose that \(a_n\geq 0\) for all \(n\). Show that \(\sum _{n=1}^\infty a_n\) is unconditionally convergent.
(Problem 2983) Let \(\sum _{n=1}^\infty a_n\) be an unconditionally convergent series that converges to \(\infty \). Suppose that \(a_n\neq \infty \) for all \(n\). Show that \(\sum _{n=1}^\infty \max (a_n,0)\) converges absolutely.
[Definition: Signed measure] If \((X,\M )\) is a measurable space, then a signed measure on \((X,\M )\) is a function \(\nu \) such that
\(\nu :\M \to [-\infty ,\infty ]\).
\(\nu (\emptyset )=0\).
If \(\{E_k\}_{k=1}^\infty \subseteq \M \) and \(E_k\cap E_j=\emptyset \) whenever \(k\neq j\), then \(\sum _{k=1}^\infty \nu (E_k)\) converges and
(Problem 2984) Suppose that \(\{E_k\}_{k=1}^\infty \subseteq \M \) and that \(E_k\cap E_j=\emptyset \) whenever \(k\neq j\). Show that \(\sum _{k=1}^\infty \nu (E_k)\) converges unconditionally.
(Problem 2985) Suppose that \(\{E_k\}_{k=1}^\infty \subseteq \M \) and that \(E_k\cap E_j=\emptyset \) whenever \(k\neq j\), and that
(Irina, Problem 3000) Let \((X,\M )\) be a measurable space and \(\nu \) a signed measure on \((X,\M )\). Suppose that \(B\in \M \) and that \(\nu (B)=\pm \infty \). Show that \(\nu (E)=\pm \infty \) for all \(E\in \M \) with \(B\subseteq E\).
(Problem 2987) Let \(\nu \) be a signed measure. Show that \(\nu \) assumes at most one of the two values \(-\infty \) and \(\infty \); that is, show that either \(\nu :\M \to [-\infty ,\infty )\) or \(\nu :\M \to (-\infty ,\infty ]\) (or both).
(Problem 2988) Let \(f:\R \to [0,\infty ]\) be measurable. Define \(\mu (A)=\int _A f\) for all measurable sets \(A\subseteq \R \). Show that \(\mu \) is a measure on the \(\sigma \)-algebra \(\M \) of Lebesgue measurable subsets of \(\R \).
(Problem 2990) Let \((X,\M )\) be a measurable space, let \(\mu :\M \to [0,\infty ]\) be a measure, and let \(\lambda :\M \to [0,\infty )\) be a finite measure (that is, a measure that satisfies \(\lambda (X)<\infty \)). Let \(\alpha \), \(\beta \in \R \) and let \(\nu =\alpha \mu +\beta \lambda \). Show that \(\nu \) is a signed measure on \((X,\M )\).
(Problem 3010) Let \((X,\M )\) be a measurable space and \(\nu \) a signed measure on \((X,\M )\). Suppose that \(E\in \M \) and that \(\nu (E)>-\infty \). Show that \(\inf \{\nu (B):B\in \M ,B\subseteq E\}>-\infty \).
(Problem 3011) Let \((X,\M )\) be a measurable space and \(\nu \) a signed measure on \((X,\M )\). Suppose that \(E\in \M \) and that \(\nu (E)<\infty \). Show that \(\sup \{\nu (B):B\in \M ,B\subseteq E\}<\infty \).
[Definition: Positive with respect to a signed measure] Let \((X,\M )\) be a measurable space and \(\nu \) a signed measure on \((X,\M )\). If \(A\in \M \), we say that \(A\) is positive with respect to \(\nu \) if \(\nu (E)\geq 0\) for every \(E\in \M \) with \(E\subseteq A\).
(Problem 3012) Let \(A\) be positive with respect to \(\nu \) and let \(\M _A=\{E\in \M :E\subseteq A\}\). Then \((A,\M _A,\nu \big \vert _{\M _A})\) is a measure space.
[Definition: Null with respect to a signed measure] Let \((X,\M )\) be a measurable space and let \(\nu \) be a signed measure on \((X,\M )\). If \(A\in \M \), we say that \(A\) is null with respect to \(\nu \) if \(\nu (E)= 0\) for every \(E\in \M \) with \(E\subseteq A\).
(Problem 3013) Give an example of a set of (signed) measure zero that is not a null set.
Proposition 17.4. Let \((X,\M )\) be a measurable space and \(\nu \) a signed measure on \((X,\M )\). Then every measurable subset of a positive set is positive and the union of countably many positive sets is positive.
(Problem 3020) Prove Proposition 17.4.
Hahn’s Lemma. Let \((X,\M )\) be a measurable space and \(\nu \) a signed measure on \((X,\M )\). Let \(E\in \M \) satisfy \(0<\nu (E)<\infty \). Then there is an \(A\in \M \) such that \(A\subseteq E\), \(A\) is positive, and \(\nu (A)>0\).
(Problem 3030) Prove Hahn’s Lemma.
(Problem 3031) Is the condition \(\nu (E)<\infty \) in Hahn’s Lemma a necessary condition?
The Hahn Decomposition Theorem. Let \((X,\M )\) be a measurable space and let \(\nu \) be a signed measure on \(\M \). Then there is a positive set \(A\) and a negative set \(B\) such that
(Problem 3040) Prove the Hahn Decomposition Theorem.
[Definition: Mutually singular measure]Let \((X,\M )\) be a measurable space and let \(\mu \), \(\lambda \) be two (nonnegative) measures on \(\M \). We say that \(\mu \perp \lambda \), or \(\mu \) and \(\lambda \) are mutually singular, if there are sets \(A\) and \(B\) in \(\M \) with \(X=A\cup B\) and with \(\mu (A)=\lambda (B)=0\).
(Problem 3041) Let \(m\) denote the Lebesgue measure on \(\R \) and let \(\delta _3\) denote the Dirac measure (that is, \(\delta _3(E)=1\) if \(3\in E\) and \(\delta _3(E)=0\) if \(3\notin E\)). Show that \(m\perp \delta _3\).
The Jordan Decomposition Theorem. Let \((X,\M )\) be a measurable space and let \(\nu \) be a signed measure on \(\M \). Then there is a unique pair of nonnegative measures \(\nu ^+\) and \(\nu ^-\) on \(\M \) that are mutually singular and such that \(\nu =\nu ^+-\nu ^-\).
(Problem 3050) Show that \(\nu ^+\) and \(\nu ^-\) exist.
[Chapter 17, Problem 13] Show that the ordered pair \((\nu ^+,\nu ^-)\) is unique.
[Definition: Measurable function] Let \((X,\M )\) be a measurable space and let \(f:X\to [-\infty ,\infty ]\). Suppose that for every \(c\in \R \) the set
Proposition 18.1. Let \((X,\M )\) be a measurable space and let \(f:X\to [-\infty ,\infty ]\). The following statements are equivalent.
Furthermore, if any of these conditions is true, then \(f^{-1}(\{c\})\) is an element of \(\M \) for all \(c\in [-\infty ,\infty ]\).
Proposition 18.2. If \(f\) is measurable, then \(f^{-1}(\mathcal {O})\) is measurable for all open sets \(\mathcal {O}\subseteq \R \).
[Chapter 18, Problem 1] If \(f\) is measurable, then \(f^{-1}(B)\) is measurable for all Borel sets \(B\subseteq \R \).
Proposition 18.3. Let \((X,\M ,\mu )\) be a complete measure space and let \(f:X\to [-\infty ,\infty ]\). Suppose that \(g:X\to [-\infty ,\infty ]\) satisfies \(f=g\) almost everywhere on \(X\) and that \(g\) is measurable. Then \(f\) is also measurable.
(Problem 3060) Prove Proposition 18.3.
[Chapter 18, Problem 2] If \((X,\M ,\mu )\) is not complete, then Proposition 18.3 is false.
Theorem 18.4. If \(f\) and \(g\) are both real-valued measurable functions on the measurable space \((X,\M )\), then so are \(\alpha f+\beta g\), \(fg\), \(\max (f,g)\), and \(\min (f,g)\) for any \(\alpha \), \(\beta \in \R \).
Theorem 2.41. (Axler) Let \(\B \) be the \(\sigma \)-algebra of Borel sets of real numbers. Suppose that \(B\subseteq \R \) is a Borel set and that \(\varphi :B\to \R \) is continuous. Then \(\varphi \) is measurable with respect to \(\B \). (We call such functions Borel measurable.)
(Problem 3061) Prove Theorem 2.41 (Axler).
Theorem 2.44. (Axler) Let \((X,\M )\) be a measurable space, let \(B\subseteq \R \) be a Borel set, let \(f:X\to B\) be measurable with respect to \(\M \), and let \(\varphi :B\to \R \) be Borel measurable. Show that \(\varphi \circ f:X\to \R \) is measurable with respect to \(\M \).
(Problem 3070) Prove Theorem 2.44 (Axler).
Theorem 2.48. (Axler) Let \((X,\M )\) be a measurable space. Let \(f_n:X\to [-\infty ,\infty ]\) be \(\M \)-measurable. Suppose that \(f:X\to [-\infty ,\infty ]\) and that \(f_n(x)\to f(x)\) for every \(x\in X\). Then \(f\) is measurable.
(Problem 3080) Prove Theorem 2.48 (Axler).
Theorem 18.6. Let \((X,\M ,\mu )\) be a complete measure space. Let \(f_n:X\to [-\infty ,\infty ]\) be measurable. Suppose that \(f:X\to [-\infty ,\infty ]\) and that \(f_n(x)\to f(x)\) for \(\mu \)-almost every \(x\in X\). Then \(f\) is measurable.
[Chapter 18, Problem 3] If \((X,\M ,\mu )\) is a measure space that is not complete, then there is a sequence \(\{f_n\}_{n=1}^\infty \) of \(\M \)-measurable functions that converge pointwise almost everywhere to a function that is not \(\mu \)-measurable.
(Problem 3090) Prove Theorem 18.6.
Corollary 18.7. Let \((X,\M )\) be a measurable space. Let \(f_n:X\to [-\infty ,\infty ]\) be \(\M \)-measurable. Then \(\sup _n f_n\), \(\inf _n f_n\), \(\liminf _n f_n\), and \(\limsup _n f_n\) are all measurable.
The simple approximation lemma. Let \((X,\M )\) be a measurable space. Let \(f:X\to \R \) be measurable and bounded. Let \(\varepsilon >0\). Then there are two simple functions \(\varphi _\varepsilon \) and \(\psi _\varepsilon \) with
The simple approximation theorem. Let \((X,\M )\) be a measurable space. Let \(f:X\to [-\infty ,\infty ]\). Then \(f\) is measurable if and only if there is a sequence \(\{\varphi _n\}_{n=1}^\infty \) such that
If \(\mu :\M \to [0,\infty ]\) is a measure and \((X,\M ,\mu )\) is \(\sigma \)-finite, then we may in addition require that \(\mu \{x\in X:\varphi _n(x)\neq 0\}<\infty \).
Egoroff’s theorem. Let \((X,\M ,\mu )\) be a measure space with \(\mu (X)<\infty \). Let \(\{f_n\}_{n=1}^\infty \) be a sequence of measurable functions on \(X\) that converges pointwise almost everywhere to some function \(f\) that is finite almost everywhere.
Then for every \(\varepsilon >0\) there is a measurable set \(F\subseteq X\) with \(m(X\setminus F)<\varepsilon \) and such that \(f_n\to f\) uniformly on \(F\).
[Axler, Definition 3.1: \(\M \)-partition] Let \(\M \) be a \(\sigma \)-algebra on a set \(X\). An \(\M \)-partition of \(X\) is a collection \(\P \) such that
\(\P \subseteq \M \),
\(\P \) is finite (that is, \(\P \) is a collection of finitely many sets),
\(X=\bigcup _{A\in \P } A\),
if \(A\), \(B\in \P \) then either \(A=B\) or \(A\cap B=\emptyset \).
[Axler, Definition 3.2: Lower Lebesgue sum] Let \((X,\M ,\mu )\) be a measure space and let \(f:X\to [0,\infty ]\) be measurable with respect to \(\M \). If \(P\subset \M \) is a \(\M \)-partition of \(X\), then
[Axler, Definition 3.3: Integral of a nonnegative measurable function] Let \((X,\M ,\mu )\) be a measure space and let \(f:X\to [0,\infty ]\) be measurable. We define
(Problem 3100) Let \((X,\M ,\mu )\) be a measure space and let \(f\), \(g:X\to [0,\infty ]\) be measurable. Suppose that \(g=f\) \(\mu \)-almost everywhere on \(X\). Show that \(\int _X f\,d\mu =\int _X g\,d\mu \).
Theorem 3.8. (Axler) Let \((X,\M ,\mu )\) be a measure space and let \(f\), \(g:X\to [0,\infty ]\) be measurable functions. Suppose that \(f(x)\leq g(x)\) for all \(x\in X\). Then \(\int _X f\,d\mu \leq \int _X g\,d\mu \).
(Problem 3110) Prove Theorem 3.8 (Axler).
Theorem 3.7. (Axler) Suppose that \(\varphi \) is a nonnegative simple function, that is, \(\varphi =\sum _{k=1}^n c_k \chi _{E_k}\) for some real numbers \(c_k\), where the \(E_k\)s are pairwise disjoint measurable sets. Then
Theorem 3.15. (Axler) Suppose that \(\varphi =\sum _{k=1}^n c_k \chi _{E_k}\) for some real numbers \(c_k\) where the \(E_k\)s are measurable sets. Then
(Problem 3120) Prove Theorem 3.7 (Axler).
Theorem 3.9. (Axler) Let \((X,\M ,\mu )\) be a measure space and let \(f:X\to [0,\infty ]\) be measurable. Then
(Problem 3130) Prove Theorem 3.9 (Axler).
Theorem 3.11. (Axler) (The monotone convergence theorem.) Let \((X,\M ,\mu )\) be a measure space, and let \(\{f_n\}_{n=1}^\infty \) be a sequence of nonnegative measurable functions \(f_n:X\to [0,\infty ]\). Suppose in addition that \(f_n(x)\leq f_{n+1}(x)\) for all \(x\in X\). Then
(Problem 3140) Prove Theorem 3.11 (Axler).
Theorem 18.11. If \((X,\M ,\mu )\) is a measure space, \(f\), \(g:X\to [0,\infty ]\) are measurable, and \(\alpha \), \(\beta >0\), then
(Problem 3150) Use Theorem 3.15 (Axler), Theorem 3.11 (Axler) (the monotone convergence theorem), and the simple approximation theorem to prove Theorem 18.11.
Fatou’s lemma. Let \((X,\M ,\mu )\) be a measure space, and let \(\{f_n\}_{n=1}^\infty \) be a sequence of nonnegative measurable functions \(f_n:X\to [0,\infty ]\). Then
[Chapter 3A, Problem 17 (Axler)] Use Theorem 3.11 (Axler) (the monotone convergence theorem) to prove Fatou’s lemma.
[Axler, Definition 3.18: Integral of a real valued function] Let \((X,\M ,\mu )\) be a measure space and let \(f:X\to [-\infty ,\infty ]\) be \(\M \)-measurable. Suppose that \(\int _X f^+\,d\mu \) and \(\int _X f^-\,d\mu \) are not both infinite. Then we define
Theorem 3.20. (Axler) Let \((X,\M ,\mu )\) be a measure space and let \(f\), \(g:X\to [-\infty ,\infty ]\) be \(\M \)-measurable. If \(\int _X f\,d\mu \) exists and \(c\in \R \), then \(\int _X cf\,d\mu =c\int _X f\,d\mu \).
Theorem 3.21. (Axler) Let \((X,\M ,\mu )\) be a measure space and let \(f\), \(g:X\to [-\infty ,\infty ]\) be \(\M \)-measurable. If \(\int _X |f|\,d\mu +\int _X |g|\,d\mu <\infty \), then \(\int _X (f+g)\,d\mu \) exists and
(Problem 3160) Prove Theorem 3.21 (Axler).
Theorem 3.22. (Axler) Let \((X,\M ,\mu )\) be a measure space and let \(f\), \(g:X\to [-\infty ,\infty ]\) be \(\M \)-measurable. Suppose that \(f(x)\leq g(x)\) for all \(x\in X\) and that \(\int _X f\,d\mu \), \(\int _X g\,d\mu \) exist. Then \(\int _X f\,d\mu \leq \int _X g\,d\mu \).
(Problem 3170) Prove Theorem 3.22 (Axler) in the case where \(\int _X |f|\,d\mu \) and \(\int _X |g|\,d\mu \) are both finite.
(Problem 3171) Prove Theorem 3.22 (Axler) in the case where \(\int _X |f|\,d\mu \) or \(\int _X |g|\,d\mu \) is infinite.
Theorem 3.23. (Axler) Let \((X,\M ,\mu )\) be a measure space and let \(f:X\to [-\infty ,\infty ]\) be \(\M \)-measurable. If \(\int _X f\,d\mu \) exists then
(Problem 3180) Prove Theorem 3.23 (Axler).
(Problem 3190) Let \((X,\M ,\mu )\) be a measure space, let \(E\in \M \), and let \(f:X\to [-\infty ,\infty ]\) be \(\M \)-measurable. Show that
[Axler, Definition 3.24: Integral over a subset] Let \((X,\M ,\mu )\) be a measure space, let \(E\in \M \), and let \(f:X\to [-\infty ,\infty ]\) be \(\M \)-measurable. We define
Theorem 3.25. (Axler) Let \((X,\M ,\mu )\) be a measure space, let \(E\in \M \), and let \(f:X\to [-\infty ,\infty ]\) be \(\M \)-measurable and such that \(\int _E f\,d\mu \) is defined. Then
(Problem 3200) Prove Theorem 3.25 (Axler).
Theorem 3.28. (Axler) Let \((X,\M ,\mu )\) be a measure space, let \(g:X\to [0,\infty ]\) be \(\M \)-measurable, and suppose that \(\int _X g\,d\mu <\infty \). Then for every \(\varepsilon >0\), there exists a \(\delta >0\) such that, if \(B\in \M \) and \(\mu (B)<\delta \), then
Theorem 3.29. (Axler) Let \((X,\M ,\mu )\) be a measure space, let \(g:X\to [0,\infty ]\) be \(\M \)-measurable, and suppose that \(\int _X g\,d\mu <\infty \). Then for every \(\varepsilon >0\), there exists an \(E\in \M \) with \(\mu (E)<\infty \) and with
Theorem 2.85. (Axler) (Egorov’s theorem.) Let \((X,\M ,\mu )\) be a measure space with \(\mu (X)<\infty \). Let \(\{f_k\}_{k=1}^\infty \) be a sequence of \(\M \)-measurable functions on \(X\) that converges pointwise almost everywhere to some function \(f\) that is finite almost everywhere.
Then for every \(\varepsilon >0\) there is a set \(E\in \M \) with \(m(X\setminus E)<\varepsilon \) and such that \(f_k\to f\) uniformly on \(E\).
Theorem 3.31. (Axler) (The Dominated Convergence Theorem). Let \((X,\M ,\mu )\) be a measure space, let \(g:X\to [0,\infty ]\) be \(\M \)-measurable, and suppose that \(\int _X g\,d\mu <\infty \).
If \(f_k:X\to [-\infty ,\infty ]\) is \(\M \)-measurable, \(|f_k(x)|\leq g(x)\) for all \(k\in \N \) and all \(x\in X\), and if \(f(x)=\lim _{k\to \infty } f_k(x)\) exists for almost every \(x\in X\), then
(Problem 3210) Prove Theorem 3.31 (Axler) in the case where \(\mu (X)<\infty \).
(Problem 3220) Prove Theorem 3.31 (Axler) in the case where \(\mu (X)=\infty \).
(Problem 3230) Let \((X,\M ,\mu )\) be a measure space, let \(f:X\to [0,\infty ]\) be \(\M \)-measurable, and let \(\{A_n\}_{n=1}^\infty \) be a sequence of pairwise-disjoint sets in \(\M \).
If \(A=\bigcup _{n=1}^\infty A_n\), show that
Theorem 18.13. Let \((X,\M ,\mu )\) be a measure space, let \(f:X\to [-\infty ,\infty ]\) be \(\M \)-measurable, and let \(\{A_n\}_{n=1}^\infty \) be a sequence of pairwise-disjoint sets in \(\M \). If at least one of \(\int _X f^+\,d\mu \) and \(\int _X f^-\,d\mu \) is finite, then
(Problem 3240) Prove Theorem 18.13.
(Problem 3241) Let \((X,\M ,\mu )\) be a measure space and let \(f:X\to [0,\infty ]\). Show that \(\int _\emptyset f\,d\mu =0\).
(Problem 3242) Let \((X,\M ,\mu )\) be a measure space and let \(f:X\to [0,\infty ]\). Define \(\nu :\M \to [0,\infty ]\) by
(Problem 3243) Let \(X\) be a mathematical model of a portion of the universe, that is, a subset of \(\R ^3\) together with certain functions that describe physical properties of the region.
Euclid is a geometer, and his favorite measure on \(X\) is \(\mu _3\), the volume (Lebesgue) measure on \(\R ^3\).
Newton is a physicist, and his favorite measure is \(\nu \), where \(\nu (E)\) is the mass of all the material in the set \(E\).
Coulomb is a physicist studying charge, and his favorite measure is \(\lambda \), where \(\lambda (E)\) is the total electrical charge in the set \(E\).
If \(D\) represents mass density (or mass per unit volume, in kilograms per cubic meter), and if \(Q\) represents the charge density (or charge per unit mass, in coulombs per kilogram), show that \(\nu (E)=\int _E D\,d\mu _3\) and \(\lambda (E)=\int _E Q\,d\nu \).
[Definition: Absolutely continuous] If \((X,\M )\) is a measurable space and \(\mu \), \(\nu \) are two measures on \((X,\M )\), we say that \(\nu \) is absolutely continuous with respect to \(\mu \), or \(\nu \ll \mu \), if for all \(E\in \M \) such that \(\mu (E)=0\) we have that \(\nu (E)=0\).
(Problem 3244) Let \((X,\M ,\mu )\) be a measure space and let \(f:X\to [0,\infty ]\). Define \(\nu :\M \to [0,\infty ]\) by
The Radon-Nikodym Theorem. Let \((X,\M )\) be a measurable space and let \(\mu \), \(\nu \) be two measures on \((X,\M )\) such that \(\nu \ll \mu \) and such that \((X,\M ,\mu )\) is \(\sigma \)-finite.
Then there is a \(\M \)-measurable function \(f:\M \to [0,\infty ]\) such that
[Chapter 18, Problem 60] Let \(X=[0,1]\) and let \(\M \) be the set of Lebesgue measurable subsets of \(X\). Let \(\mu \) denote the counting measure and let \(\nu \) denote the Lebesgue measure. Then \(\nu \ll \mu \) but there is no \(\M \)-measurable function \(f:X\to [0,\infty ]\) that satisfies the conclusions of the Radon-Nikodym theorem.
(Problem 3260) (This is the first step in the proof of the Radon-Nikodym theorem.) Let \((X,\M )\) be a measurable space and let \(\mu \), \(\nu \) be two measures on \((X,\M )\) such that \(\nu \ll \mu \). Define
(Problem 3270) Let \(f\in \mathcal {F}\). Suppose that \(\int _X f\,d\mu =\nu (X)<\infty \). Show that \(\int _E f\,d\mu =\nu (E)\) for all \(E\in \M \).
(Problem 3280) Let \((X,\M )\) be a measurable space and let \(\mu \), \(\theta \) be two measures on \((X,\M )\) such that \(\theta \ll \mu \) and such that \(0<\theta (X)\) and \(\mu (X)<\infty \). Show that there exists a function \(\varphi \) such that
\(\varphi :X\to [0,\infty )\),
\(\varphi \) is \(\M \)-measurable,
\(\int _E \varphi \,d\mu \leq \theta (E)\) for all \(E\in \M \),
\(\int _X \varphi \,d\mu >0\).
(Problem 3290) Prove the Radon-Nikodym Theorem in the case where \(\mu (X)<\infty \) and \(\nu (X)<\infty \).
[Homework 27.1] Show that the Radon-Nikodym theorem is true if \(\mu (X)<\infty \), whether \(\nu \) is finite, \(\sigma \)-finite, or not \(\sigma \)-finite.
[Chapter 18, Problem 49] If the Radon-Nikodym theorem is true whenever \(\mu (X)<\infty \) and \(\nu (X)<\infty \), then it is true whenever \((X,\M ,\mu )\) and \((X,\M ,\nu )\) are \(\sigma \)-finite.
(Problem 3291) Show that the Radon-Nikodym theorem is true if \((X,\M ,\mu )\) is \(\sigma \)-finite, whether \(\nu \) is finite, \(\sigma \)-finite, or not \(\sigma \)-finite.
(Problem 3292) Show that the function \(f\) in the Radon-Nikodym theorem is unique up to redefinition on sets of measure zero.
[Definition: Radon-Nikodym derivative] The function \(f\) in the Radon-Nikodym theorem is called the Radon-Nikodym derivative of \(\nu \) with respect to \(\mu \) and is often denoted \(\frac {d\nu }{d\mu }\).
Corollary 18.20. Let \((X,\M ,\mu )\) be a \(\sigma \)-finite measure space and let \(\nu :\M \to [-\infty ,\infty ]\) be a signed measure on \(\M \) such that \(\nu \ll \mu \).
Then there is a \(\M \)-measurable function \(f:\M \to [0,\infty ]\) such that at most one of \(\int _X f^+\,d\mu \) and \(\int _X f^-\,d\mu \) is infinite and such that
(We say that \(\nu \ll \mu \) if, whenever \(\mu (E)=0\), we have that \(E\) is a null set for \(\nu \).)
Recall [Problem 6.38a]: Let \([a,b]\subset \R \) be a closed and bounded interval and let \(f:[a,b]\to \R \) be a function.
We say that \(f\) is absolutely continuous if, for every \(\varepsilon >0\), there is a \(\delta >0\) such that, if \(\{a_k\}_{k=1}^\infty \) and \(\{b_k\}_{k=1}^\infty \) satisfy
\(a\leq a_k\leq b_k\leq b\) for all \(1\leq k\leq \infty \),
\((a_k,b_k)\cap (a_j,b_j)=\emptyset \) if \(j\neq k\),
\(\sum _{k=1}^\infty |b_k-a_k|<\delta \),
then
Recall [Theorem 6.8]: If \(f\) is absolutely continuous, then \(f=g-h\), where \(g\) and \(h\) are both absolutely continuous and nondecreasing.
(Problem 3293) Let \(X=[a,b]\subset \R \) be a closed bounded interval, let \(\mathcal {B}\) be the Borel subsets of \(X\), and let \(\nu \) be a finite measure on \(\B \). Let \(g:X\to \R \) be given by \(g(x)=\nu ([a,x])\). Show that \(g\) is a nondecreasing function.
(Problem 3294) Let \(\nu \) and \(g\) be as in the previous problem. Show that \(g\) is right continuous, that is, \(\lim _{x\to c^+} g(x)=g(c)\) for all \(c\in [a,b)\).
(Problem 3300) Let \(\nu \) and \(g\) be as in the previous problem. Let \(c\in (a,b]\). Show that \(\nu ([a,c))=g(c-)\), where \(g(x\pm )\) is as in Problem 1671. Conclude that \(\nu (\{c\})=0\) if and only if \(g\) is continuous at \(c\).
Proposition AB.1. Let \(g:[a,b]\to \R \) be a nondecreasing function. Then there is a unique measure \(\nu \) on the collection of Borel subsets of \([a,b]\) such that
\(\nu ([a,x])=g(x+)-g(a)\) for all \(x\in [a,b]\),
\(\nu ([a,x))=g(x-)-g(a)\) for all \(x\in (a,b]\)
where \(g(x\pm )\) is as in Problem 1671.
In particular, if \(g(a)=0\) then \(g(x+)=\nu ([a,x])\) for all \(x\in [a,b]\), and so if \(g(a)=0\) and \(g\) is right continuous everywhere in \([a,b)\) then \(g\) is the function given by Problem 3293.
Proposition AB.2. Let \([a,b]\subset \mathbb {R}\) be a closed and bounded interval. Let \(g:[a,b]\to \R \) be nondecreasing and let \(\nu \) be as in the previous proposition.
Then \(\nu \) is absolutely continuous with respect to the Lebesgue measure if and only if \(g\) is an absolutely continuous function.
(Problem 3301) In this problem we begin the proof of Proposition AB.1. Let \(\S =\mathcal {I}\cup \mathcal {P}\), where \(\mathcal {P}=\{\{c\}:c\in [a,b]\}\) and where \(\mathcal {I}\) is the collection of all open intervals that are subsets of \([a,b]\). Show that \(\S \) is a semiring.
(Problem 3310) Let \(g:[a,b]\to \R \) be nondecreasing. Recall the definition of \(g(x\pm )\) of Problem 1671. Define \(\bar \nu :\S \to [0,\infty )\) by (for \(a\leq \alpha <\beta \leq b\) and \(c\in (a,b)\))
\(\bar \nu ((\alpha ,\beta ))=g(\beta -)-g(\alpha +)\),
\(\bar \nu (\{a\})=g(a+)-g(a)\),
\(\bar \nu (\{b\})=g(b)-g(b-)\),
\(\bar \nu (\{c\})=g(c+)-g(c-)\).
Show that, if \(I\), \(J\), and \(I\cup J\in S\), then \(\bar \nu (I\cup J)\geq \bar \nu (I)+\bar \nu (J)\).
(Problem 3320) Let \(g\) and \(\bar \nu \) be as in Problem 3310. If \(I\in \S \) and \(I=\bigsqcup _{k=1}^n I_k\) for some \(I_k\in \S \) with \(I_j\cap I_k=\emptyset \) when \(j\neq k\), show that \(\bar \nu (I)=\sum _{k=1}^n \bar \nu (I_k)\).
(Problem 3321) Let \(g\) and \(\bar \nu \) be as in Problem 3310. Show that \(\bar \nu \) is a premeasure on \(\S \).
(Problem 3322) Prove Proposition AB.1. That is, let \(g\) and \(\bar \nu \) be as in Problem 3310. Show that the induced Carathéodory measure \(\nu \) is a Borel measure (that is, all Borel sets are measurable) and that \(\nu \) is an extesion of \(\bar \nu \).
Recall [Theorem 3.28 (Axler)]: Let \((X,\M ,\mu )\) be a measure space, let \(f:X\to [0,\infty ]\) be \(\M \)-measurable, and suppose that \(\int _X f\,d\mu <\infty \). Then for every \(\varepsilon >0\), there exists a \(\delta >0\) such that, if \(E\in \M \) and \(\mu (E)<\delta \), then
Proposition 18.19. Let \((X,\M )\) be a measurable space and let \(\mu \), \(\nu \) be two measures on \((X,\M )\) such that \(\nu (X)<\infty \). Then \(\nu \ll \mu \) if and only if, for every \(\varepsilon >0\), there is a \(\delta >0\) such that, if \(E\in \M \) and \(\mu (E)<\delta \), then \(\nu (E)<\varepsilon \).
(Problem 3323) Assume that for every \(\varepsilon >0\), there is a \(\delta >0\) such that, if \(E\in \M \) and \(\mu (E)<\delta \), then \(\nu (E)<\varepsilon \). Show that \(\nu \ll \mu \).
(Problem 3324) Prove Proposition 18.19 in the case where \((X,\M ,\mu )\) is \(\sigma \)-finite (and thus the Radon-Nikodym theorem is true).
(Irina, Problem 3250) Prove Proposition 18.19 in the case where \((X,\M ,\mu )\) is not \(\sigma \)-finite (and so the Radon-Nikodym theorem is not available).
(Problem 3330) Let \(g:[a,b]\to \R \) be absolutely continuous and nondecreasing. Let \(\nu \) be as in Problem 3310. Show that \(\nu \) is absolutely continuous with respect to the Lebesgue measure.
(Problem 3340) Let \(\nu \) be a measure on the Borel subsets of \([a,b]\subset \R \) and let \(g(x)=\nu ([a,x])\). Assume that \(\nu \) is absolutely continuous with respect to the Lebesgue measure. Show that \(g\) is absolutely continous.
(Problem 3350) Let \(\nu \) be a measure on the Borel subsets of \([a,b]\subset \R \) and let \(g(x)=\nu ([a,x])\). Assume that \(\nu \) is absolutely continuous with respect to the Lebesgue measure (or, equivalently, that \(g\) is an absolutely continuous function). By Lebesgue’s theorem, the derivative \(g'\) exists almost everywhere. Show that \(g'=\frac {d\nu }{dm}\), where \(m\) denotes Lebesgue measure.
Recall [mutually singular measure]: If \((X,\M )\) is a measurable space and \(\mu \), \(\nu \) are two (nonnegative) measures on \((X,\M )\), we say that \(\mu \) and \(\nu \) are mutually singular, or \(\nu \perp \mu \), if there are sets \(A\in \M \) and \(B\in \M \) such that \(X=A\cup B\), \(\nu (A)=0\), and \(\mu (B)=0\).
(Problem 3351) Show that we may require \(A\) and \(B\) to be disjoint.
(Problem 3352) Show that the Lebesgue measure and the Dirac measure are mutually singular.
Recall: If \(\nu \) is a signed measure, then the measures \(\nu ^+\) and \(\nu ^-\) in the Jordan Decomposition Theorem are mutually singular.
The Lebesgue Decomposition Theorem. Let \((X,\M )\) be a measurable space and let \(\nu \) and \(\mu \) be two measures on \(\M \). Suppose that \((X,\M ,\nu )\) is \(\sigma \)-finite.
Then there exists a unique pair of measures \(\nu _0\) and \(\nu _1\) on \(\M \) such that \(\nu _0\perp \mu \), \(\nu _1\ll \mu \), and such that \(\nu (E)=\nu _0(E)+\nu _1(E)\) for all \(E\in \mathcal {M}\).
(Problem 3353) In this problem we begin the proof of the Lebesgue Decomposition Theorem. Let
Show that (for \(k=0\) and \(k=1\)),
\(\emptyset \in \S _k\),
\(\S _k\subseteq \M \),
(Problem 3360) Show that (for \(k=0\) and \(k=1\)),
If \(A\in \S _k\) and \(E\in \M \), then \(A\cap E\in \S _k\),
If \(\{A_n\}_{n=1}^\infty \subseteq \S _k\) then \(\bigcup _{n=1}^\infty A_n\in \S _k\).
(Because \(\S _0\) and \(\S _1\) are not closed under complements, they are not \(\sigma \)-algebras.)
(Problem 3361) Define \(\nu _k\) by
Show that
\(\nu _k(\emptyset )=0\),
\(0\leq \nu _k(E)\leq \nu (E)\) for all \(E\in \M \),
If \(D\subseteq E\) then \(\nu _k(D)\leq \nu _k(E)\).
(Problem 3370) Let \(\{E_n\}_{n=1}^\infty \) be a sequence of sets in \(\M \). Show that \(\nu _k\Bigl (\bigcup _{n=1}^\infty E_n\Bigr )\leq \sum _{n=1}^\infty \nu _k(E_n)\).
(Problem 3380) Let \(\{E_n\}_{n=1}^\infty \) be a sequence of pairwise-disjoint sets in \(\M \). Show that \(\nu _k\Bigl (\bigcup _{n=1}^\infty E_n\Bigr )\geq \sum _{n=1}^\infty \nu _k(E_n)\).
(Problem 3390) If \(E\in \M \), show that there is an \(A_k\in \S _k\) with \(A_k\subseteq E\) and with \(\nu _k(E)=\nu (A_k)=\nu _k(A_k)\).
(Problem 3391) Suppose that \(A\in \S _0\cap S_1\). Show that \(\nu (A)=0\).
(Problem 3400) If \(E\in \M \), show that \(\nu _0(E)+\nu _1(E)\leq \nu (E)\).
(Problem 3410) If \(E\in \M \), show that \(\nu _0(E)+\nu _1(E)\geq \nu (E)\).
(Problem 3420) Show that \(\nu _1\ll \mu \).
(Problem 3430) Suppose that \(\nu (X)<\infty \). Show that \(\nu _0\perp \mu \).
(Problem 3440) Suppose that \((X,\M ,\nu )\) is \(\sigma \)-finite. Show that \(\nu _0\perp \mu \). (This completes the proof of the existence in the Lebesgue decomposition theorem.)
[Homework 27.3] Prove the uniqueness in the Lebesgue decomposition theorem. That is, let \((X,\M )\) be a measurable space and let \(\mu \) and \(\nu \) be two measures on \(\M \). Suppose further that there are four measures \(\nu _0\), \(\nu _1\), \(\tilde \nu _0\), and \(\tilde \nu _1\) on \(\M \) such that
\(\nu =\nu _0+\nu _1=\tilde \nu _0+\tilde \nu _1\),
\(\nu _0\perp \mu \) and \(\tilde \nu _0\perp \mu \),
\(\nu _1\ll \mu \) and \(\tilde \nu _1\ll \mu \).
Show that \(\nu _0=\tilde \nu _0\) and \(\nu _1=\tilde \nu _1\). Do not assume that any of the measures involved are \(\sigma \)-finite.
[Homework 28.1] Give an example showing that existence can fail if \((X,\M ,\nu )\) is not \(\sigma \)-finite.
(Problem 3450) Let \(\S \) be a semiring of subsets of a set \(X\) and let \(\bar \mu :\S \to [0,\infty ]\) be a premeasure. Let \(\mu ^*\) be the induced Carathéodory outer measure on \(2^X\) and let \(\M \) be the \(\sigma \)-algebra of \(\mu ^*\)-measurable subsets of \(X\), and let \(\mu \) be the induced Carathéodory measure. Let \(\mathcal {B}\) be the smallest \(\sigma \)-algebra of subsets of \(X\) with \(\S \subseteq \B \). If \(\mu (X)<\infty \), show that \((X,\M ,\mu )\) is the completion of \((X,\B ,\mu \big \vert _\B )\) (where the completion is as in Problem 2700).
(Problem 3451) Show that the previous result still holds if \(\mu (X)=\infty \) but \((X,\B ,\mu )\) (or \((X,\S ,\bar \mu )\)) is \(\sigma \)-finite.
[Definition: Measurable rectangle] Let \((X,\mathcal {A},\mu )\) and \((Y,\mathcal {B},\nu )\) be two measure spaces. If \(A\in \mathcal {A}\), \(B\in \mathcal {B}\), \(\mu (A)<\infty \), and \(\nu (B)<\infty \), we call the Cartesian product \(A\times B\subseteq X\times Y\) a measurable rectangle.
Proposition 20.2a. If \(\mathcal {R}\) denotes the collection of measurable rectangles in \(X\times Y\), then \(\mathcal {R}\) is a semiring.
(Problem 3460) Prove Proposition 20.2a.
Lemma 20.1. Suppose that \(A\times B\) is a measurable rectangle and that
(Problem 3470) Prove Lemma 20.1.
(Problem 3471) Let \(\S \) be a semiring and \(\{E_n\}_{n=1}^\infty \) a sequence of elements of \(\S \). Then there is a sequence \(\{D_n\}_{n=1}^\infty \) with
\(\bigcup _{n=1}^\infty D_n=\bigcup _{n=1}^\infty E_n\),
\(D_m\cap D_n=\emptyset \) if \(m\neq n\),
for each \(n\) there exists a \(k\) with \(D_n\subseteq E_k\),
Proposition 20.2b. Let \(\mathcal {R}\) be the collection of measurable rectangles in \(X\times Y\) and let \(\bar \lambda :\mathcal {R}\to [0,\infty ]\) be given by
(Problem 3480) Prove Proposition 20.2b.
[Definition: Product measure] Let \((X,\mathcal {A},\mu )\) and \((Y,\mathcal {B},\nu )\) be two measure spaces, let \(\mathcal {R}=\{A\times B:A\in \mathcal {A},B\in \mathcal {B}\}\) be the collection of measurable rectangles in \(A\times B\), and let \(\lambda :\mathcal {R}\to [0,\infty ]\) be given by \(\lambda (A\times B)=\mu (A)\times \nu (B)\).
The product measure \(\mu \times \nu \) is the Carathéodory measure induced by \(\lambda \).
We will let \(\mathcal {A}\otimes \mathcal {B}\) denote the smallest \(\sigma \)-algebra containing \(\mathcal {R}\), and will let \(\mathcal {A}\circledast \mathcal {B}\) denote the \(\sigma \)-algebra of \((\mu \times \nu )^*\)-measurable sets; recall from Problem 3451 that if \((X\times Y,\mathcal {A}\otimes \mathcal {B},\mu \times \nu )\) is \(\sigma \)-finite then \(\mathcal {A}\circledast \mathcal {B}\) is the completion of \(\mathcal {A}\otimes \mathcal {B}\).
(Problem 3481) Show that \((\mu \times \nu )(A\times B)=\mu (A)\times \nu (B)\) for all \(A\in \mathcal {A}\) and \(B\in \mathcal {B}\).
(Problem 3482) If \((X,\mathcal {A},\mu )\) and \((Y,\mathcal {B},\nu )\) are both \(\sigma \)-finite, show that \((X,\mathcal {A}\circledast \mathcal {B},\mu \times \nu )\) is also \(\sigma \)-finite.
(Problem 3483) Let \((X,\mathcal {A},\mu )\) and \((Y,\mathcal {B},\nu )\) be two measure spaces. Define \(\phi :X\times Y\to Y\times X\) by \(\phi ((x,y))=(y,x)\). Show that \(E\in \A \otimes \B \) if and only if \(\phi (E)\in \B \otimes \A \), where \(\phi (E)=\{\phi ((x,y)):(x,y)\in E\}\).
(Micah, Problem 3500) If \(E\subseteq X\times Y\), show that \((\mu \times \nu )^*(E)=(\nu \times \mu )^*(\phi (E))\).
(Problem 3485) Show that \(E\in \A \circledast \B \) if and only if \(\phi (E)\in \B \circledast \A \).
(Muhammad, Problem 3510) If \(f:X\times Y\to [0,\infty ]\) is nonnegative and either \((\mathcal {A}\times \mathcal {B})\)-measurable or \((\mathcal {A}\circledast \mathcal {B})\)-measurable, show that \(f\circ \phi \) is \((\mathcal {B}\times \mathcal {A})\)-measurable or \((\mathcal {B}\circledast \mathcal {A})\)-measurable, respectively, and that
Proposition 20.15. Let \(n\) and \(k\) be two integers. Let \(\phi :\R ^n\times \R ^k\to \R ^{n+k}\) be given by \(\phi (((x_1,\dots ,x_n),(y_1,y_k)))=(x_1,\dots ,x_n,y_1,y_k)\). \(\phi \) is clearly a bijection.
Then:
(Problem 3490) If \(E\subseteq \R ^n\times \R ^k\), show that \((\mu _n\times \mu _k)^*(E)\leq \mu _{n+k}^*(\phi (E))\).
(Problem 3491) Prove part (a) of Proposition 20.15.
(Problem 3492) Prove Proposition 20.15.
Fubini’s theorem. Let \((X,\mathcal {A},\mu )\) and \((Y,\mathcal {B},\nu )\) be two measure spaces.
Let \(f:X\times Y\to [-\infty ,\infty ]\). Assume that either
\(f\) is \((\mathcal {A}\otimes \mathcal {B})\)-measurable, or
\(f\) is \((\mathcal {A}\circledast \mathcal {B})\)-measurable and \((Y,\mathcal {B},\nu )\) is complete.
If \(\int _{X\times Y} |f|\,d(\mu \times \nu )<\infty \), then
Tonelli’s theorem. Let \((X,\mathcal {A},\mu )\) and \((Y,\mathcal {B},\nu )\) be two \(\sigma \)-finite measure spaces.
Let \(f:X\times Y\to [0,\infty ]\). Assume that either
\(f\) is \((\mathcal {A}\otimes \mathcal {B})\)-measurable, or
\(f\) is \((\mathcal {A}\circledast \mathcal {B})\)-measurable, and \((Y,\mathcal {B},\nu )\) is complete.
If \(f\) is nonnegative, then
Corollary 20.7. Let \((X,\mathcal {A},\mu )\) and \((Y,\mathcal {B},\nu )\) be two \(\sigma \)-finite measure spaces.
Let \(f:X\times Y\to [-\infty ,\infty ]\). Assume that either
\(f\) is \((\mathcal {A}\otimes \mathcal {B})\)-measurable, or
\(f\) is \((\mathcal {A}\circledast \mathcal {B})\)-measurable and \((X,\mathcal {A},\mu )\) and \((Y,\mathcal {B},\nu )\) are both complete.
If
(Problem 3520) Assume that Tonelli’s theorem and Fubini’s theorem are both correct. Prove Corollary 20.7.
[Chapter 20, Problem 5] Let \((X,\mathcal {A},\mu )=(Y,\B ,\nu )=(\N ,2^\N ,c)\) where \(c\) is the counting measure. Let \(f:X\times Y\to [0,\infty )\) be given by
Proposition 20.5. [This is the first step in the proof of Fubini’s and Tonelli’s theorems.] Let \(E\in \mathcal {A}\circledast \mathcal {B}\). Suppose that either \(E\in \mathcal {A}\otimes \mathcal {B}\) or \((Y,\mathcal {B},\nu )\) is complete.
If \(x\in X\), define
Then
(Problem 3521) If \(F\), \(G\in X\times Y\), show that \(F_x\cup G_x=(F\cup G)_x\).
(Problem 3522) If \(F\), \(G\in X\times Y\) with \(F\cap G=\emptyset \), show that \(F_x\cap G_x=\emptyset \).
(Problem 3530) Prove Proposition 20.5 in the case where \(E\in \mathcal {R}\) (that is, where \(E=A\times B\) is a measurable rectangle).
(Problem 3540) Prove part (a) of Proposition 20.5. (Hint: Let \(\S \) be the collection of subsets of \(X\times Y\) for which the conclusion of part (a) is true. Show that \(\S \) is a \(\sigma \)-algebra and also contains \(\mathcal {R}\).)
(Problem 3541) Let \(\mathcal {W}\) be the collection of all \(E\subseteq X\times Y\) such that part (b) is true, that is, such that \(\Psi _E\) is measurable. Give an example to show that \(\mathcal {W}\) may not be a \(\sigma \)-algebra.
(Problem 3571) Prove Proposition 20.5 in the case where \(E\) is a \(\mathcal {R}_\sigma \) set.
(Problem 3572) Prove that the intersection of two \(\mathcal {R}_\sigma \) sets is a \(\mathcal {R}_\sigma \) set.
(Problem 3573) Prove Proposition 20.5 in the case where \(E\) is a \(\mathcal {R}_{\sigma \delta }\) set.
(Problem 3574) Prove Proposition 20.5 in the case where \(\mu \times \nu (E)=0\).
(Problem 3575) Prove Proposition 20.5.
Theorem 20.6. Let \((X,\mathcal {A},\mu )\) and \((Y,\B ,\nu )\) be two \(\sigma \)-finite measure spaces. Let \(\varphi :X\times Y\to [0,\infty )\) be a \(\mathcal {A}\circledast \mathcal {B}\)-measurable simple function that is integrable. Suppose that either \(\varphi \) is \(\mathcal {A}\otimes \mathcal {B}\)-measurable or that \((Y,\mathcal {B},\nu )\) is complete. Then the conclusion of Fubini’s theorem holds:
For \(\mu \)-almost every \(x\in X\) we have that \(\varphi _x\) is \(\mathcal {B}\)-measurable and integrable over \(Y\).
The function \(\Phi (x)=\int _Y \varphi _x\,d\nu \) is \(\A \)-measurable and integrable over \(X\).
\(\int _{X\times Y} \varphi \,d(\mu \times \nu )=\int _X \Bigl [\int _Y \varphi (x,y)\,d\nu (y)\Bigr ]d\mu (x).\)
(Problem 3580) Prove Theorem 20.6 in the case where \(\varphi =\chi _E\), where \(E\in \mathcal {A}\circledast \B \) with \((\mu \times \nu )(E)<\infty \).
(Problem 3590) Show that if the conclusion of Fubini’s theorem holds for \(f\) and \(g\), then it holds for \(\alpha f+\beta g\) for all \(\alpha \), \(\beta \in \R \). (This completes the proof of Theorem 20.6.)
(Problem 3600) Prove Tonelli’s theorem.
(Problem 3610) Prove Fubini’s theorem in the case where \((X,\mathcal {A},\mu )\) and \((Y,\B ,\nu )\) are \(\sigma \)-finite.
(Problem 3640) Suppose that \((X,\mathcal {A},\mu )\) and \((Y,\mathcal {B},\nu )\) are two measure spaces that are not necessarily \(\sigma \)-finite. Let \(f:X\times Y\to [0,\infty ]\) be \(\mathcal {A}\circledast \B \)-measurable and integrable.
Show that
for some \(A\in \mathcal {A}\) and \(B\in \B \) such that \((A,\mathcal {A}_A,\mu _A)\) and \((B,\mathcal {B}_B,\nu _B)\) (as defined in Problem 17.6) are both \(\sigma \)-finite. Hint: Apply Problem 2760 to \(\{(x,y):|f(x,y)|>1/n\}\).
(Problem 3650) Prove that Fubini’s theorem is true even if \((X,\mathcal {A},\mu )\) and \((Y,\mathcal {B},\nu )\) are not \(\sigma \)-finite.
[Definition: \(L^p(X,\mu )\)] If \((X,\M ,\mu )\) is a measure space and \(0<p<\infty \), then \(L^p(X,\mu )\) is the set of all equivalence classes of \(\M \)-measurable functions \(f:X\to [-\infty ,\infty ]\), where the equivalence relation is \(f\sim g\) if \(f=g\) \(\mu \)-almost everywhere (that is, \(f=g\) on \(E\), where \(E\subseteq X\) satisfies \(\mu (X\setminus E)=0\)), such that some (hence every) representative \(f\) of the equivalence class satisfies
Theorem 19. If \((X,\M ,\mu )\) is a measure space and \(1\leq p\leq \infty \), then Hölder’s inequality, Minkowski’s Inequality, and the Riesz-Fischer Theorem are true with \(L^p(E)\) replaced by \(L^p(X,\mu )\). If \(1\leq p<\infty \), then the Riesz representation theorem for the dual of \(L^p(E)\) is true with \(L^p(E)\) replaced by \(L^p(X,\mu )\). If \(1<p<\infty \), then every bounded sequence in \(L^p(X,\mu )\) has a weakly convergent subsequence and the Banach-Saks theorem is true with \(L^p(E)\) replaced by \(L^p(X,\mu )\).
(Problem 3660) Let \((X,\M ,\mu )\) be a measure space. Recall that \(L^\infty (X,\mu )\) is a space of equivalence classes of functions with the equivalence relation given by \(f\equiv g\) if \(f=g\) \(\mu \)-almost everywhere in \(X\). Show that any element \(F\) of \(L^\infty (X,\mu )\) must have a representative \(f:X\to \R \) that is bounded and satisfies \(\sup _X |f|=\|F\|_{L^\infty (X,\mu )}\).
[Definition: Finitely additive signed measure] Let \((X,\M )\) be a measurable space. Let \(\nu :\M \to [-\infty ,\infty ]\). Suppose that
\(\nu (\emptyset )=0\).
If \(A\), \(B\in \M \) with \(A\cap B=\emptyset \), then \(\nu (A)+\nu (B)\) is defined and equals \(\nu (A\cup B)\).
Then \(\nu \) is called a finitely additive signed measure.
(Problem 3661) A finitely additive signed measure cannot assume both of the two values \(\infty \) and \(-\infty \).
[Definition: Bounded finitely additive signed measure] If \(\nu \) is a finitely additive signed measure on \((X,\M )\), then the total variation of \(\nu \) over \(E\in \M \), denoted \(|\nu |(E)\), is
[Definition: Integral of a simple function with respect to a bounded finitely additive signed measure] Let \((X,\M )\) be a measurable space, let \(\varphi :X\to \R \) be a \(\M \)-measurable simple function, and let \(\nu \) be a bounded finitely additive signed measure on \((X,\M )\). Then
(Problem 3663) Show that, if \(\varphi \) and \(\psi \) are two \(\M \)-measurable simple functions and \(\alpha \), \(\beta \in \R \), then
[Definition: Integral of a bounded function with respect to a bounded finitely additive signed measure] Let \((X,\M )\) be a measurable space, let \(f:X\to \R \) be \(\M \)-measurable and bounded, and let \(\nu \) be a bounded finitely additive signed measure on \((X,\M )\). If \(\{\varphi _n\}_{n=1}^\infty \) is a sequence of bounded \(\M \)-measurable simple functions with \(\varphi _n\to f\) uniformly on \(X\), define
Begin the proof that \(\int _X f\,d\nu \) is well defined by showing that some such sequence \(\{\varphi _n\}_{n=1}^\infty \) exists.
Continue the proof that \(\int _X f\,d\nu \) is well defined by showing that if \(\varphi _n\to f\) uniformly on \(X\), and if each \(\varphi _n\) is a bounded simple \(\M \)-measurable function, then \(\lim _{n\to \infty } \int _X\varphi _n \,d\nu \) exists.
(Problem 3680) Complete the proof that \(\int _X f\,d\nu \) is well defined by showing that if \(\varphi _n\to f\) and \(\psi _n\to f\), uniformly on \(X\), and if each \(\varphi _n\) and \(\psi _n\) is a bounded simple \(\M \)-measurable function, then
(Problem 3681) Show that, if \(f\) and \(g\) are two \(\M \)-measurable bounded functions and \(\alpha \), \(\beta \in \R \), then
[Definition: Absolute continuity of finitely additive signed measures] Let \((X,\M )\) be a measurable space, let \(\mu \) be a measure on \((X,\M )\), and let \(\nu \) be a finitely additive signed measure on \((X,\M )\). We say that \(\nu \ll \mu \) if, whenever \(E\in \M \) and \(\mu (E)=0\), we have that \(\nu (E)=0\).
(Problem 3683) Show that if \(\nu \ll \mu \) and \(\mu (E)=0\), then \(|\nu |(E)=0\).
[Definition: Integral of an element of \(L^\infty (X,d\mu )\) with respect to a bounded finitely additive signed measure] Let \((X,\M ,\mu )\) be a measure space and let \(\nu \) be a bounded finitely additive signed measure with \(\nu \ll \mu \). If \(F\in L^\infty (X,\mu )\), then we define
(Problem 3690) At least one such \(f\) must exist by Problem 3660. Show that \(\int _X f\,d\nu \) is well defined by showing that, if \(f\), \(g\in F\) and \(f\) and \(g\) are both bounded, then \(\int _X f\,d\nu =\int _X g\,d\nu \).
(Problem 3692) Show that, if \(F\) and \(G\) are two elements of \(L^\infty (X,\mu )\) and \(\alpha \), \(\beta \in \R \), then
[Definition: The space of bounded finitely additive absolutely continuous signed measures] If \((X,\M ,\mu )\) is a measure space, then \(\mathcal {BFA}(X,\M ,\mu )\) is the set of all bounded finitely additive signed measures on \((X,\M )\) that are absolutely continuous with respect to \(\mu \). This is a vector space under the representation
Theorem 19.7. (The Kantorovitch Representation Theorem.) Let \((X,\M ,\mu )\) be a measure space. The mapping \(T:\mathcal {BFA}(X,\M ,\mu )\to (L^\infty (X,\mu ))^*\) given by
(Problem 3693) Show that \(T_\nu \in (L^\infty (X,\mu ))'\) for all \(\nu \in \mathcal {BFA}(X,\M ,\mu )\) and that \(\|T_\nu \| \leq |\nu |(X)\).
(Problem 3700) Prove the Kantorovitch Representation Theorem by showing that \(\|T_\nu \| \geq |\nu |(X)\) and that, if \(T\in (L^\infty (X,\mu ))^*\), then there is a \(\nu \in \mathcal {BFA}(X,\M ,\mu )\) such that \(T=T_\nu \).
(Memory 3701) If \((X,d)\) and \((Y,\varrho )\) are metric spaces and \(f:X\to Y\) is continuous, then \(f^{-1}(G)\) is open in \((X,d)\) whenever \(G\) is open in \((Y,\varrho )\), and \(f^{-1}(F)\) is closed in \((X,d)\) whenever \(F\) is closed in \((Y,\varrho )\).
(Memory 3702) If \((X,d)\) and \((Y,\varrho )\) are two metric spaces, if \(X=D\cup E\), if \(D\) and \(E\) are both open, and if \(f:X\to Y\) is such that \(f\big \vert _D\) and \(f\big \vert _E\) are both continuous, then \(f\) is continuous.
[Definition: Compact set] If \((X,d)\) is a metric space and \(K\subseteq X\), then we say that \(K\) is compact if, whenever \(\S \) is a collection of open subsets of \(X\) and \(K\subseteq \bigcup _{G\in \S } G\), there is a finite subset \(\S _0\subseteq \S \) with \(K\subseteq \bigcup _{G\in \S _0} G\).
(Memory 3703) Every compact set in a metric space is closed.
(Problem 3704) Every closed subset of a compact set is compact.
[Definition: Support] Let \((X,d)\) be a metric space and let \(f:X\to \R \). Then the support of \(f\), denoted \(\supp f\), is the closure in \((X,d)\) of \(\{x\in X:f(x)\neq 0\}\). (Thus \(x\in X\setminus \supp f\) if and only if \(x\in X\) and there is an \(r>0\) such that \(f(y)=0\) for all \(y\in B(x,r)\).)
[Definition: Compact support] Let \((X,d)\) be a metric space and let \(f:X\to \R \). We say that \(f\) has compact support if \(\supp f\) is compact.
(Memory 3705) [Urysohn’s lemma] If \((X,d)\) is a metric space, \(F\subseteq G\subseteq X\), \(F\) is closed, and \(G\) is open, then there is a continuous function \(f:X\to [0,1]\) with \(f(x)=1\) for all \(x\in F\) and \(f(x)=0\) for all \(x\notin G\).
[Definition: Locally compact metric space] A metric space \((X,d)\) is locally compact if, for every \(x\in X\), there exists an \(r>0\) such that \(\overline B(x,r)\) is compact.
(Problem 3706) If \((X,\M ,\mu )\) is a measure space and \(\M \) contains a sequence of countably many pairwised disjoint sets of positive measure, show that \(L^p(X,\mu )\) is not locally compact.
Proposition 21.2. If \((X,d)\) is a locally compact metric space and \(K\subseteq U\subseteq X\), where \(K\) is compact and \(U\) is open, then there is a set \(V\) with \(K\subseteq V\subseteq \overline V\subseteq U\) and such that \(V\) is open and \(\overline V\) is compact.
(Bonus Problem 3707) Prove Proposition 21.2.
[Definition: \(C_c(X)\)] Let \((X,d)\) be a locally compact metric space and let \(C_c(X)\) be the set of all compactly supported continuous functions \(f:X\to \R \). Then \(C_c(X)\) is a vector space. We can make \(C_c(X)\) into a normed vector space by imposing the norm \(\|f\|=\sup _X |f|\).
[Definition: Positive linear functional] Let \(T:C_c(X)\to \R \) be a bounded linear functional. We say that \(T\) is positive if \(T(f)\geq 0\) whenever \(f(x)\geq 0\) for all \(x\in X\).
[Definition: Borel measure] If \((X,d)\) is a metric space and \(\B \) is the smallest \(\sigma \)-algebra over \(X\) that contains all open sets, then we call \(\mu \) a Borel measure if
the domain of \(\mu \) is \(\B \),
if \(K\subseteq X\) is compact then \(\mu (K)<\infty \).
(Problem 3708) Let \((X,d)\) be a metric space and let \(\B \) be the smallest \(\sigma \)-algebra over \(X\) that contains all open sets. If \(f:X\to \R \) is continuous, show that \(f\) is \(\B \)-measurable.
(Problem 3709) Let \((X,d)\) be a metric space and let \(\B \) be the smallest \(\sigma \)-algebra over \(X\) that contains all open sets. Let \(\mu :\B \to [0,\infty ]\) be a Borel measure. If \(f:X\to \R \) is continuous and bounded, define \(T_\mu (f)=\int _X f\,d\mu \). Show that \(T_\mu \) is a positive bounded linear operator from \(C_c(X)\) to \(\R \).
[Definition: Radon measure] If \((X,d)\) is a locally compact metric space, then we say that \(\mu \) is a Radon measure on \(X\) if
\(\mu \) is a Borel measure,
if \(E\in \B \) then \(\mu (E)=\inf \{\mu (U):U\) open, \(E\subseteq U\}\),
if \(G\) is open then \(\mu (G)=\sup \{\mu (K):K\subseteq G\) is compact\(\}\).
Proposition 21.11. Suppose that \((X,d)\) is a locally compact metric space, \(\B \) is the smallest \(\sigma \)-algebra over \(X\) that contains the open sets, and \(\mu \) and \(\nu \) are two Radon measures on \(\B \). Supppose that
(Problem 3710) Prove Proposition 21.11.
The Riesz-Markov theorem. If \((X,d)\) is a locally compact metric space, and if \(T:C_c(X)\to \R \) is a positive bounded linear functional, then there is at least one Borel measure \(\mu \) on \(X\) such that
(Problem 3711) In this problem we begin the proof of the Riesz-Markov theorem. If \(\mathcal {O}\subseteq X\) is open, define
Proposition 21.5. Let \(\{U_k\}_{k=1}^n\) be a collection of finitely many (possibly overlapping) open sets in a metric space \((X,d)\). Let \(K\subseteq \bigcup _{k=1}^n U_k\) be compact.
Then there is a collection of continuous functions \(\varphi _k:X\to [0,1]\) such that \(\varphi _k=0\) outside of \(U_k\) and \(\varphi =\sum _{k=1}^n\varphi _k\) satisfies \(\varphi (x)=1\) if \(x\in K\) and \(\varphi (x)\in [0,1]\) if \(x\in X\).
(Bonus Problem 3712) Prove Proposition 21.5.
(Problem 3720) Show that \(\bar \mu \) is countably subadditive.
[Homework 22.1] The outer measure \(\mu ^*\) induced by the set function \(\bar \mu \) satisfies \(\mu ^*(\mathcal {O})=\bar \mu (\mathcal {O})\) for every \(\mathcal {O}\) in the domain of \(\bar \mu \) if and only if \(\bar \mu \) is countably subadditive.
(Problem 3730) Show that \(\mu ^*\) is a Carathéodory outer measure. What does Theorem 20.28 imply about \(\mu ^*\) and the induced Carathéodory measure \(\mu \)?
(Problem 3740) Show that \(\mu \) is a Borel measure (that is, if \(K\subseteq X\) is compact then \(\mu (K)<\infty \)).
(Problem 3750) Show that \(\mu \) is a Radon measure (that is, if \(E\in \B \) then \(\mu (E)=\inf \{\mu (U):U\) open, \(E\subseteq U\}\), and if \(G\) is open then \(\mu (G)=\sup \{\mu (K):K\subseteq G\) is compact\(\}\).
(Problem 3760) Show that \(T(f)=\int _X f\,d\mu \) for all nonnegative \(f\in C_c(X)\).