[Definition: Field] A field is a set \(\bf F\) along with operations \(+\) and \(\cdot \) with the following properties:
(Problem 10) Show that if \(a\), \(b\), \(c\in {\bf F}\) satisfy \(a+b=a+c=0\) then \(b=c\).
(Problem 20) Show that if \(a\), \(b\), \(c\in {\bf F}\) satisfy \(ab=ac=1\) then \(b=c\).
(Problem 30) Suppose that \(\bf F\) is a field. Show that if \(a\in {\bf F}\) then \(0a=0\).
[Definition: Inverses] If \(\bf F\) is a field and \(a\in {\bf F}\), then:
(Problem 40) Show that \((-(-a))=a\).
(Problem 50) Suppose that \(\bf F\) is a field and that \(a\in {\bf F}\). Show that \(a(-1)=(-a)1=(-a)\).
(Problem 60) Suppose that \(\bf F\) is a field and that \(a\in {\bf F}\). Show that \(aa=(-a)(-a)\).
[Definition: Ordered field] An ordered field is a field \(\bf F\) together with a subset \(P\subset {\bf F}\), called the positive subset, such that:
(Problem 70) Let \(\bf F\) be an ordered field with positive subset \(P\). Show that \(1\in P\).
(Problem 80) Let \(a\in {\bf F}\) with \(a\neq 0\). Show that \(a\in P\) if and only if \(a^{-1}\in P\).
[Definition: Comparison] Let \(\bf F\) be an ordered field with positive subset \(P\). Let \(a\), \(b\in {\bf F}\).
(Problem 90) Show that the relation \(<\) is transitive. Show that the relation \(\leq \) is reflexive. Is either relation symmetric?
(Problem 100) Let \(a\), \(b\), \(c\in {\bf F}\). Suppose that \(a\leq b\). Show that \(a+c\leq b+c\).
(Problem 110) Let \(\bf F\) be an ordered field. Let \(a\), \(b\in {\bf F}\). Show that exactly one of the following is true:
[Definition: Inductive set] Let \(\bf F\) be an ordered field. A subset \(I\subseteq {\bf F}\) is inductive if:
(Problem 120) Let \(\bf F\) be an ordered field. Give three examples of inductive subsets of \(\bf F\).
(Problem 130) Let \(\bf F\) be an ordered field. Suppose that \(\mathcal {I}\) is a collection of inductive subsets of \(F\) (that is, if \(J\in \mathcal {I}\) then \(J\subseteq {\bf F}\) and \(J\) is inductive). Show that \(\bigcap _{J\in \mathcal {I}} J\) is inductive.
(Problem 140) Let \(\bf F\) be an ordered field. Let \(N\) be the intersection of all inductive subsets of \(\bf F\). (Thus \(N\) is an inductive set.) Let \(m\in N\). Show that \(1\leq m\).
(Problem 150) Let \(m\), \(k\in N\). Show that \(m+k\in N\).
(Problem 170) Let \(k\in N\). Show that there does not exist a \(m\in N\) with \(k<m<k+1\).
(Problem 160) Let \(m\), \(k\in N\) with \(k>m\). Show that \(k-m\in N\).
(Problem 180) Let \(\bf F\) and \(\bf \widetilde F\) be two ordered fields. Let \(N\), \(\widetilde N\) be the intersection of all inductive subsets of \(\bf F\) and \(\bf \widetilde F\), respectively. If \(m\in N\), let \(J_m=\{n\in N:n\leq m\}\). Suppose that there are two numbers \(n\), \(k\in N\) (not necessarily distinct) and two functions \(\varphi _k\), \(\varphi _n\) that satisfy
Show that \(\varphi _n(i)=\varphi _k(i)\) for all \(i\in J_k\cap J_n\).
(Problem 190) Show that if \(m\in N\) then a function \(\varphi _m:J_m\to \widetilde N\) as in Problem 180 must exist.
(Problem 200) Let \(\mathbf {F}\) and \(\mathbf {\widetilde F}\) be two ordered fields, with \(\mathbf {N}\), \(\mathbf {\widetilde N}\) the respective minimal inductive sets. Let \(\varphi :\mathbf {N}\to \mathbf {\widetilde N}\) be given by \(\varphi (n)=\varphi _n(n)\) for all \(n\in N\). Let \(\widetilde \varphi \) be defined as \(\varphi \) with the roles of \(\bf R\) and \(\bf \widetilde R\) reversed. Show that
(Problem 210) We will define the natural numbers (up to isomorphism) as a minimal inductive set of an ordered field. Let \(\bf F\) be an ordered field. Define subsets \(Z\) and \(Q\) of \(\bf F\) that we expect to be isomorphic to the integers \(\mathbf {Z}\) and rational numbers \(\mathbf {Q}\).
[Definition: Upper and lower bounds] Let \(\bf F\) be an ordered field and let \(A\subseteq {\bf F}\). We say that \(\ell \in {\bf F}\) is a lower bound for \(A\) if \(\ell \leq a\) for all \(a\in A\). We say that \(u\) is an upper bound for \(A\) if \(u\geq a\) for all \(a\in A\).
[Definition: Least upper bound] Let \(\bf F\) be an ordered field and let \(A\subseteq F\). We say that \(b\in F\) is the least upper bound for \(A\) if
[Definition: Greatest lower bound] Let \(\bf F\) be an ordered field and let \(A\subseteq F\). We say that \(b\in F\) is the greatest lower bound for \(A\) if
(Problem 220) Let \(\bf F\) be an ordered field and let \(A\subseteq F\). Show that \(A\) has at most one least upper bound.
(Problem 230) Let \(\bf F\) be an ordered field and let \(A\subseteq F\). Suppose that \(a\in A\) is an upper bound for \(A\). Show that \(a\) is a least upper bound.
(Problem 240) Let \(A\subseteq N\) where \(N\) is as in Problem 140. Suppose that \(A\) is nonempty and that \(A\) has an upper bound in \(N\). Show that \(A\) has a least upper bound.
(Problem 250) Let \(A\subseteq N\) where \(N\) is as in Problem 140. Suppose that \(A\) has a least upper bound. Show that the least upper bound is an element of \(A\).
[Definition: Completeness] An ordered field \(\bf F\) is complete if every nonempty subset of \(\bf F\) with an upper bound has a least upper bound.
(Problem 251) Let \(F\) be a complete ordered field and let \(N\) be as in Problem 140. Show that \(N\) has no upper bound.
(Problem 252) Let \(F\) be a complete ordered field, let \(r\in F\), and let \(s\in F\) with \(s>0\). Show that there is a \(n\in N\) with \(n>r\) and an \(m\in N\) with \(1/m<s\).
[Definition: The real numbers] The real numbers \(\mathbf {R}\) are a complete ordered field. (We saw in MATH 4513 that a complete ordered field exists. You can review the argument in Section 0B.)
(Problem 260) Let \(\mathbf {F}\) and \(\mathbf {\widetilde F}\) be two ordered fields, with \(\mathbf {N}\), \(\mathbf {\widetilde N}\) the respective minimal inductive sets. Let \(\varphi :\mathbf {N}\to \mathbf {\widetilde N}\) be the isomorphism in Problem 200. Let \(\widetilde \varphi \) be defined as \(\varphi \) with the roles of \(\bf F\) and \(\bf \widetilde F\) reversed. Show that
(Problem 270) Let \(\mathbf {Z}\), \(\mathbf {\widetilde Z}\) be as in Problem 210. Extend \(\varphi \) to a function \(\varphi :\mathbf {Z}\to \mathbf {\widetilde Z}\).
(Problem 290) Let \(\mathbf {Q}\), \(\mathbf {\widetilde Q}\) be as in Problem 210. Extend \(\varphi \) to a function \(\varphi :\mathbf {Q}\to \mathbf {\widetilde Q}\).
(Problem 301) Let \(\psi (r)=\sup \{\varphi (q):q\in \mathbf {Q},q<r\}\). Show that \(\psi :F\to \widetilde F\) is well defined and that \(\psi (q)=\varphi (q)\) for all \(q\in Q\).
[Definition: Open cover] Let \(K\subseteq \mathbf {R}\). A cover of \(K\) is a collection \(\mathcal {U}\) of subsets of \(\mathbf {R}\) that satisfies \(K\subseteq \bigcup _{V\in \mathcal {U}} V\). We say that \(\mathcal {U}\) is an open cover if every element \(V\) of \(\mathcal {U}\) is an open set in \(\mathbf {R}\).
[Definition: Finite subcover] Let \(K\subseteq \mathbf {R}\) and let \(\mathcal {U}\) be a cover of \(K\). A subcover of \(\mathcal {U}\) is any subcollection \(\mathcal {U}_1\subseteq \mathcal {U}\) such that \(K\subseteq \bigcup _{V\in \mathcal {U}_1} V\). A finite subcover is a subcover that is also a collection of finitely many sets.
[Definition: Compact set] A set \(K\subseteq \mathbf {R}\) is compact if every open cover of \(K\) has a finite subcover.
(Problem 320) State the Heine-Borel theorem.
(Problem 330) Let \(K\subseteq \mathbf {R}\) be nonempty, closed, and bounded. Show that \(\inf K\in K\) and \(\sup K\in K\).
(Problem 340) Let \(K\subseteq \mathbf {R}\) be nonempty, closed, and bounded. Let \(\mathcal {U}\) be an open cover of \(K\). Let \(K_r=\{x\in K:x\leq r\}\). Then \(K_r\subseteq K\) so \(\mathcal {U}\) is also an open cover of \(K_r\). Let \(\mathcal {S}=\{r\in K:\)there is a finite subcover of \(K_r\}\). Show that \(\mathcal {S}\) is nonempty and bounded.
(Problem 350) Let \(s=\sup \mathcal {S}\). Show that \(s=\sup K\).
(Problem 360) (The Heine-Borel theorem.) Let \(K\subset \mathbf {R}\). Suppose that \(K\) is both closed and bounded. Show that \(K\) is compact.
[Definition: Continuous function] A function \(f:X\to \bf R\), where \(X\) is a metric space, is continuous if, for every \(x\in X\) and every \(\varepsilon >0\), there is a \(\delta >0\) depending on \(x\) and \(\varepsilon \) such that if \(|x-y|<\delta \) then \(|f(x)-f(y)|<\varepsilon \). \(f\) is uniformly continuous if \(\delta \) may be chosen independent of \(x\).
(Problem 370) Let \(X\) be compact and let \(f:X\to \mathbf {R}\) be continuous. Show that \(f\) is bounded.
(Problem 380) Show that \(f\) is uniformly continuous.
(Problem 420) Show that \(\mathbf {Q}\cap [0,1]\) is countable. That is, show that there is a sequence \(\{q_k\}_{k=1}^\infty \) that contains each rational number in \([0,1]\) exactly once and contains no other numbers.
[Definition: Partition] Suppose \(a\), \(b\in \mathbf {R}\) with \(a<b\). We say that \(P\) is a partition of \([a,b]\) if:
We will write \(P=\{x_0,x_1,\dots ,x_n\}\) with \(a=x_0<x_1<\dots <x_n=b\).
[Definition: \(\sup _A\), \(\inf _A\)] If \(A\) is a set and \(f:A\to \mathbf {R}\), then \(\sup _A f=\sup \{f(x):x\in A\}\) and \(\inf _A f=\inf \{f(x):x\in A\}\).
[Definition: Lower and upper Darboux sums] Let \([a,b]\subseteq \mathbf {R}\), \(f:[a,b]\to \mathbf {R}\), and let \(P\) be a partition of \([a,b]\). Suppose that \(f\) is bounded on \([a,b]\), so \(-\infty <\inf _{[a,b]}f \leq \sup _{[a,b]}f<\infty \).
Then the upper and lower Darboux sums of \(f\) with respect to the partition \(P\) are
[Definition: Lower and upper Darboux integrals] Let \([a,b]\subseteq \mathbf {R}\) be a closed bounded interval and let \(f:[a,b]\to \mathbf {R}\) be a bounded function. We define \begin {equation*} L(f,[a,b])=\sup _P L(f,P),\qquad U(f,[a,b]) = \sup _P U(f,P) \end {equation*} where the supremum and infimum are over all partitions \(P\) of \([a,b]\).
[Definition: Riemann integral] Let \([a,b]\subseteq \mathbf {R}\) be a closed bounded interval and let \(f:[a,b]\to \mathbf {R}\) be a bounded function. If \(U(f,[a,b])=L(f,[a,b])\), then we say that \(f\) is Riemann integrable on \([a,b]\) and write \begin {equation*} \int _a^b f=U(f,[a,b])=L(f,[a,b]). \end {equation*}
(Problem 390) Let \(f(x)=\frac {1}{\root 3 \of x}\). Let \(P\) be any partition of \([0,1]\). What is \(U(f,P)\)?
(Problem 400) How did you define \(\int _0^1 \frac {1}{\root 3 \of x}dx\) in calculus? In undergraduate analysis?
(Problem 410) Can you write down a generalization of the Riemann integral that can be used to define \(\int _0^1 \frac {1}{\root 3\of {\sin \pi /x}}\,dx\)?
(Problem 440) Let \(f:[0,1]\to \bf R\). Suppose that \(f\) has a singularity at every rational number in \([0,1]\). Can we generalize the approach in Problem 410 to define \(\int _{0}^1 f\)?
(Problem 450) Let \(\xi \in (0,1)\). Let \(\{q_k\}_{k=1}^\infty \) be as in Problem 420. Write down a formula for a function \(h_k\) that is continuous, nonnegative, integrates to \(2^{-k}\xi \), and satisfies \(h_k(x)=0\) if \(|x-q_k|>2^{-k}\xi \). Let \(f_n(x)=\sum _{k=1}^n h_k(x)\). What is \(\int _{-1}^2 f_n\)? What can you say about \(\int _0^1 f_n\)?
(Problem 460) Consider the semi-metric1 space \((X,d)\), where \(X\) is the set of all bounded Riemann integrable functions on \([0,1]\), and where \(d(f,g)=\int _{0}^1 |f-g|\). Show that \(\{f_n\}_{n=1}^\infty \) is a Cauchy sequence in \(X\).
(Problem 470) Suppose (for the sake of contradiction) that \(\{f_n\}_{n=1}^\infty \) is convergent, that is, that there is a Riemann integrable function \(f\) such that \(f_n\to f\) (that is, \(\int _{0}^1 |f_n-f|\to 0\) as \(n\to \infty \)). Provide an upper bound on \(\int _{0}^1 f\).
(Problem 480) Let \((x_{j-1},x_j)\subseteq [0,1]\) with \(x_{j-1}<x_j\). Show that there is a point \(x\) in \((x_{j-1},x_j)\) such that \(f(x)\geq 1/2\).
(Problem 490) Show that \(U(f,P)\geq 1/2\) for any partition \(P\) of \([0,1]\). Have we derived a contradiction?
[Definition: Extended real number] An extended real number is either a real number, \(\infty =+\infty \), or \(-\infty \).
[Definition: Ordering on the extended real numbers] Let \(a\), \(b\) be extended real numbers. We say that \(a\leq b\) if:
(Problem 500) Let \(E\) be a set of extended real numbers. What is \(\sup E\)? What is \(\inf E\)?
(Problem 510) Write down a rigorous definition of the expression \(a+b\) where \(a\) and \(b\) are extended real numbers. Are there any pairs of values we do not want to allow?
(Problem 520) Write down a rigorous definition of the expression \(\sum _{k=0}^\infty a_k\) where the \(a_k\)s are nonnegative extended real numbers.
[Definition: Length of an open interval] Let \(I\subseteq \mathbf {R}\) be an open interval. The length of \(I\), or \(\ell (I)\), is defined to be \begin {equation*} \ell (I)=\begin {cases}b-a&\text {if $I=(a,b)$ with $-\infty <a<b<\infty $,}\\ 0&\text {if $I=\emptyset $,}\\ \infty &\text {if $I$ is unbounded.}\end {cases} \end {equation*}
[Definition: Outer measure] Let \(A\subseteq \mathbf {R}\). The outer measure \(|A|\) of \(A\) is defined to be
(Problem 580) Let \(A\subseteq \mathbf {R}\) be a countable set. Show that \(|A|=0\).
(Problem 530) Let \([a,b]\subseteq \mathbf {R}\) be a closed bounded interval. Show that \(|[a,b]|=b-a\).
(Problem 540) Let \(f\) be a bounded function defined on the closed and bounded interval \([a,b]\). Show that
(Problem 550) Show that if \(A\subseteq B\subseteq \mathbf {R}\) then \(|A|\leq |B|\).
(Problem 560) Let \(I\) be an open interval. Show that \(\ell (I)=|I|\).
(Problem 570) Let \(I=[a,b)\) or \(I=(a,b]\) be a bounded half-open interval. Show that \(|I|=b-a\).
(Problem 590) Use the previous result to show that \([a,b]\) is uncountable whenever \(a<b\).
(Problem 600) Show that if \(A\subseteq \mathbf {R}\) and \(B\subseteq \mathbf {R}\) then \(|A\cup B|\leq |A|+|B|\).
(Problem 610) Show that if \(A_k\subseteq \mathbf {R}\) for all \(k\geq 0\) then \(\Bigl |\bigcup _{k=1}^\infty A_k\Bigr |\leq \sum _{k=1}^\infty |A_k|\).
(Problem 620) Show that if \(A\) and \(B\) are disjoint sets, and if \(\sup A<\inf B\), then \(|A\cup B|=|A|+|B|\).
(Problem 630) Show that if \(A\) and \(B\) are disjoint sets, and if \(\sup A\leq \inf B\), then \(|A\cup B|=|A|+|B|\).
(Problem 640) Let \(A\subseteq \mathbf {R}\), \(t\in \mathbf {R}\), and let \(A+t=\{a+t:a\in A\}\). Show that \(|A+t|=|A|\).
[Axiom of Choice] The axiom of choice states that, if \(\mathcal {E}\) is a collection of sets, and if the elements of \(\mathcal {E}\) are pairwise-disjoint nonempty sets, there is a set \(V\) such that \(V\) contains exactly one element of each set in \(\mathcal {E}\) and no other elements.
(Problem 650) Define the relation \(\sim \) by \(a\sim b\) if \(a-b\in \mathbf {Q}\). If \(r\in [-1,1]\), let \(E_r=\{s\in [-1,1]:r\sim s\}\). Show that if \(r\), \(s\in [-1,1]\), then either \(E_r=E_s\) or \(E_r\cap E_s=\emptyset \).
(Problem 660) Let \(\mathcal {E}=\{E_r:r\in [-1,1]\}\). Let \(V=V_0\) be the set given by the axiom of choice. Show that if \(v\), \(w\in V\) with \(v\neq w\) then \(v-w\notin \mathbf {Q}\).
(Problem 670) Let \(q\) be a rational number in \([-2,2]\). Let \(V_q=\{v+q:v\in V\}\). Show that \(|V_q|=|V|\).
(Problem 680) Show that if \(q\), \(p\in \mathbf {Q}\cap [-2,2]\) then either \(q=p\) or \(V_q\cap V_p=\emptyset \).
(Problem 690) Show that \begin {equation*} [-1,1]\subseteq \bigcup _{q\in [-2,2]\cap {\bf Q}} V_q\subseteq [-3,3]. \end {equation*}
(Problem 700) Show that \(|V|>0\).
(Problem 710) 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 \(\sum _{k=1}^\infty |V_{q_k}| \neq \Bigl |\bigcup _{k=1}^\infty V_{q_k}\Bigr |\).
(Problem 720) Show that there exist two disjoint sets \(A\) and \(B\) such that \(|A\cup B| \neq |A|+|B|\).
(Problem 730) Show that if \(V\subseteq \mathbf {R}\) and \(V\) is open then \(V=\bigcup _{j=1}^\infty I_j\) for some sequence \(\{I_j\}_{j=1}^\infty \) of bounded open intervals.
(Problem 731) Let \((X,d)\) be a measure space and let \(Y\subseteq X\). Recall that \(G\subseteq Y\) is relatively open (or open in \(Y\)) if, for every \(g\in G\), there is a \(r_g>0\) such that if \(d(g,y)<r_g\) and \(y\in Y\), then \(y\in G\). Show that \(G\) is relatively open if and only if \(G=U\cap Y\) for some \(U\subseteq X\) that is open (in \(X\)).
[Definition: Inverse image] Let \(f:X\to Y\) be a function and let \(A\subseteq Y\). Then \(f^{-1}(A)=\{x\in X:f(x)\in A\}\).
(Problem 740) Let \(X\subseteq \mathbf {R}\) and let \(f:X\to \mathbf {R}\). Show that \(f\) is continuous if and only if \(f^{-1}(U)\) is open in \(X\) whenever \(U\) is open in \(\R \).
(Problem 750) Let \(f:X\to Y\) be a function and let \(A\subseteq Y\). Show \(f^{-1}(Y\setminus A)=X\setminus f^{-1}(A)\).
(Problem 760) Let \(\mathcal {B}\subseteq 2^Y\) be a collection of subsets of \(Y\). Show that \(f^{-1}\Bigl (\bigcup _{A\in \mathcal {B}} A\Bigr ) = \bigcup _{A\in \mathcal {B}} f^{-1}(A)\).
(Problem 770) Let \(\mathcal {B}\subseteq 2^Y\) be a collection of subsets of \(Y\). Show that \(f^{-1}\Bigl (\bigcap _{A\in \mathcal {B}} A\Bigr ) = \bigcap _{A\in \mathcal {B}} f^{-1}(A)\).
(Problem 780) Let \(f:X\to Y\) and \(g:Y\to Z\). Let \(A\subseteq Z\). Show that \((g\circ f)^{-1}(A)=f^{-1}(g^{-1}(A))\).
[Definition: Increasing function] Let \(X\subseteq \mathbf {R}\) and let \(f:X\to \mathbf {R}\).
(Problem 790) What properties of the outer measure \(|\,\cdot \,|\) did we use in the proof of Problems 650–720?
[Definition: \(\sigma \)-algebra; measure] Let \(X\) be a set. Let \(\mathcal {\mathcal {S}}\) be a collection of subsets of \(\mathcal {\mathcal {S}}\). Let \(\mu :\mathcal {\mathcal {S}}\to [0,\infty ]\). We say that \((X,\mathcal {\mathcal {S}})\) is a measurable space and \(\mu \) is a measure on \((X,\mathcal {\mathcal {S}})\) if:
We call the elements of \(\mathcal {\mathcal {S}}\) measurable sets.
(Problem 800) Is \(\mu (E)=|E|\) a measure on \((\mathbf {R},2^{\bf R})\), where \(2^{\bf R}\) is the collection of all subsets of \(\mathbf {R}\)?
(Problem 810) Does there exist a measure \(\mu \) on \((\mathbf {R},2^{\bf R})\) that satisfies
(Problem 820) Let \(X\) be a nonempty set. Show that \(\mathcal {\mathcal {S}}=\{\emptyset ,X\}\) is a \(\sigma \)-algebra on \(X\).
(Problem 830) Let \(X\) be a nonempty set. Show that \(\mathcal {\mathcal {S}}=2^X\) is a \(\sigma \)-algebra on \(X\).
(Problem 840) Let \(X\) be a nonempty set. Let \(C=\{E\subseteq X:E\) is countable\(\}\). Let \(\mathcal {\mathcal {S}}=C\cup \{E\subseteq X:X\setminus E\in C\}\). Show that \(\mathcal {S}\) is a \(\sigma \)-algebra on \(X\).
(Problem 850) Let \((X,\mathcal {S})\) be a measurable space (that is, let \(X\) be a set and let \(\mathcal {S}\) be a \(\sigma \)-algebra on \(X\)). Show that \(X\in \mathcal {S}\).
(Problem 860) Let \((X,\mathcal {S})\) be a measurable space. Show that if \(D\), \(E\in \mathcal {S}\) then \(D\cup E\in \mathcal {S}\).
(Problem 870) Show \(D\cap E\in \mathcal {S}\).
(Problem 880) Show \(D\setminus E\in \mathcal {S}\).
(Problem 890) Let \((X,\mathcal {S})\) be a measurable space. Let \(\{E_k\}_{k=1}^\infty \subseteq \mathcal {S}\). Show that \(\bigcap _{k=1}^\infty E_k\in \mathcal {S}\).
[Exercise 2B.11] Let \((Y,\mathcal {T})\) be a measurable space. Let \(X\in \mathcal {T}\) and let \(\mathcal {\mathcal {S}}=\{E\in \mathcal {T}:E\subseteq X\}\). Then \((X,\mathcal {\mathcal {S}})\) is a measurable space.
(Problem 900) Let \(X\) be a set. Let \(\mathcal {T}\) be a collection of \(\sigma \)-algebras on \(X\). Show that \(\bigcap _{\mathcal {S}\in \mathcal {T}} \mathcal {S}\) is also a \(\sigma \)-algebra on \(X\).
[Definition: Smallest \(\sigma \)-algebra] Let \(X\) be a set and let \(\mathcal {A}\subseteq 2^X\) be a collection of subsets of \(\mathcal {S}\). The intersection of all \(\sigma \)-algebras containing \(\mathcal {A}\) is the smallest \(\sigma \)-algebra containing \(\mathcal {A}\).
(Problem 910) Let \(X=\mathbf {R}\) and let \(\mathcal {A}=\{\{3\},\{5\}\}\). What is the smallest \(\sigma \)-algebra containing \(\mathcal {A}\)?
(Problem 920) Let \(X=\mathbf {R}\) and let \(\mathcal {A}=\{\{x\}:x\in \mathbf {R}\}\). What is the smallest \(\sigma \)-algebra containing \(\mathcal {A}\)?
[Definition: Borel set] A set \(E\subseteq \mathbf {R}\) is called a Borel set if \(E\) is in the smallest \(\sigma \)-algebra on \(\mathbf {R}\) that contains all the open subsets of \(\mathbf {R}\).
(Problem 930) Show that all closed subsets of \(\mathbf {R}\) are Borel sets.
(Problem 940) Show that all countable subsets of \(\mathbf {R}\) are Borel sets.
(Problem 950) Let \(-\infty <a<b<\infty \). Show that \([a,b)\) is a Borel set.
[Definition: Measurable function] Let \((X,\mathcal {S})\) be a measurable space. Let \(f:X\to \mathbf {R}\). We say that \(f\) is \(\mathcal {S}\)-measurable if \(f^{-1}(B)\in \mathcal {S}\) for every Borel set \(B\).
(Problem 960) Show that any function \(f:X\to \mathbf {R}\) is \(2^X\)-measurable.
(Problem 980) Let \(\mathcal {S}=\{\emptyset ,X\}\). Suppose that \(f:X\to \mathbf {R}\) is \(\mathcal {S}\)-measurable. Show that \(f\) is constant.
(Problem 970) Let \((X,\mathcal {S})\) be a measurable space and let \(f:X\to \mathbf {R}\) be a constant function. Show that \(f\) is \(\mathcal {S}\)-measurable.
(Problem 981) Let \(X\) be a set and let \(\mathcal {S}\), \(\mathcal {T}\) be two \(\sigma \)-algebras on \(X\). Suppose that \(\mathcal {T}\subseteq \mathcal {S}\) and that \(f:X\to \mathbf {R}\) is \(\mathcal {T}\)-measureable. Show that \(f\) is \(\mathcal {S}\)-measurable.
(Problem 982) Let \(Y\) be a set, let \(\mathcal {T}\) be a \(\sigma \)-algebra on \(Y\), and let \(f:Y\to \R \) be \(\mathcal {T}\)-measurable. Let \( X\in \mathcal {T}\) and let \(\mathcal {S}\) be as in Exercise 2B.11. Show that \(f\big \vert _{X}\) is \(\mathcal {S}\)-measurable.
[Definition: Characteristic function] Let \(E\subseteq X\). Then \(\chi _E:X\to \mathbf {R}\) is the piecewise defined function given by \begin {equation*} \chi _E(x)=\begin {cases}1,&x\in E,\\0,&x\notin E.\end {cases} \end {equation*}
(Problem 1000) Show that \(\chi _E\) is \(\mathcal {S}\)-measurable if and only if \(E\in \mathcal {S}\).
(Problem 1010) Let \((X,\mathcal {S})\) be a measurable space and let \(f:X\to \mathbf {R}\). Suppose that \(f^{-1}((a,\infty ))\in \mathcal {S}\) for all \(a\in \mathbf {R}\). Let \(\mathcal {T}=\{A\subseteq \mathbf {R}:f^{-1}(A)\in \mathcal {S}\}\). Show that \(\mathcal {T}\) is a \(\sigma \)-algebra on \(\mathbf {R}\).
(Problem 1020) Let \(-\infty < a<b<\infty \). Show that \((a,b)\in \mathcal {T}\).
(Problem 1030) Let \(U\subseteq \mathbf {R}\) be open. Show that \(U\in \mathcal {T}\).
(Problem 1040) Is \(f\) measurable?
(Problem 1041) Let \((X,\mathcal {S})\) be a measurable space. Let \(f:X\to \R \) be \(\mathcal {S}\)-measurable. Let \(a\in \R \). Show that \(af\) is also \(\mathcal {S}\)-measurable.
[Definition: Borel measurable] If \(X\subseteq \mathbf {R}\), then \(f:X\to \mathbf {R}\) is Borel measurable if \(f^{-1}(B)\) is a Borel set for every Borel set \(B\subseteq \mathbf {R}\). That is, \(f\) is Borel measurable if \(f\) is \(\mathcal {B}\)-measurable where \(\mathcal {B}\) is the set of all Borel subsets of \(\R \).
(Problem 1042) Let \(f:X\to \R \) be Borel measurable. Show that \(X\) is Borel.
(Problem 1050) Let \(f:\mathbf {R}\to \mathbf {R}\) be continuous. Show that \(f\) is Borel measurable.
(Problem 1060) Let \(X\subseteq \mathbf {R}\) be a Borel set and let \(f:X\to \mathbf {R}\) be continuous. Show that \(f\) is Borel measurable.
(Problem 1070) Let \(X\subseteq \mathbf {R}\) be a Borel set and let \(f:X\to \mathbf {R}\) be increasing. Show that \(f\) is Borel measurable.
(Problem 1080) Let \((X,\mathcal {S})\) be a measurable space. Let \(Y\subseteq \mathbf {R}\). Let \(f:X\to Y\) be \(\mathcal {S}\)-measurable and let \(g:Y\to \mathbf {R}\) be Borel measurable. Show that \(g\circ f:X\to \mathbf {R}\) is \(\mathcal {S}\)-measurable.
(Problem 1081) Let \(B\) be a Borel set. Let \(a\in \R \). Show that \(aB=\{ab:b\in B\}\) is Borel.
(Problem 1090) Let \((X,\mathcal {S})\) be a measurable space. Let \(f\), \(g:X\to \mathbf {R}\) be \(\mathcal {S}\)-measurable functions. Show that \(f+g\) is \(\mathcal {S}\)-measurable.
(Problem 1100) Show that \(fg\) is \(\mathcal {S}\)-measurable.
(Problem 1110) If \(g(x)\neq 0\) for all \(x\in X\), show that \(f/g\) is \(\mathcal {S}\)-measurable.
(Problem 1120) Let \((X,\mathcal {S})\) be a measurable space and let \(\{f_n\}_{n=1}^\infty \) be a sequence of \(\mathcal {S}\)-measurable functions \(f_n:X\to \mathbf {R}\). Suppose that \(f(x)=\lim _{n\to \infty } f_n(x)\) exists for each \(x\in X\). Show that \(f\) is \(\mathcal {S}\)-measurable.
[Definition: Borel subsets of the extended real numbers] Let \(B\subseteq [-\infty ,\infty ]\). We say that \(B\) is a Borel set if \(B\cap \mathbf {R}\) is a Borel set. If \((X,\mathcal {S})\) is a measurable space and \(f:X\to [-\infty ,\infty ]\), we say that \(f\) is \(\mathcal {S}\)-measureable if \(f^{-1}(B)\in \mathcal {S}\) for every Borel set \(B\subseteq [-\infty ,\infty ]\).
(Problem 1121) Let \((X,\mathcal {S})\) be a measure space and let \(f:X\to [-\infty ,\infty ]\). Let \(\widetilde X=f^{-1}(\R )\) and let \(\widetilde {\mathcal {S}}=\{E\in \mathcal {S}:E\subseteq X\}\). Show that \(f\) is \(\mathcal {S}\)-measurable if and only if
(Problem 1130) Suppose \((X,\mathcal {S})\) is a measurable space and \(f:X\to [-\infty ,\infty ]\) satisfies \(f^{-1}((a,\infty ])\in \mathcal {S}\) for all \(a\in \mathbf {R}\). Show that \(f\) is \(\mathcal {S}\)-measurable.
(Problem 1140) Let \((X,\mathcal {S})\) be a measurable space. Let \(f_n:X\to [-\infty ,\infty ]\) be \(\mathcal {S}\)-measurable for each \(n\). Show that \(g(x)=\sup \{f_n(x):n\in \mathbf {N}\}\) is \(\mathcal {S}\)-measurable.
(Problem 1150) Show that \(h(x)=\inf \{f_n(x):n\in \mathbf {N}\}\) is \(\mathcal {S}\)-measurable.
(Problem 1151) If \(\{f_i\}_{i\in \mathcal {I}}\) is a collection of measurable functions, and \(\mathcal {I}\) is an uncountable index set, must \(g(x)=\sup \{f_i(x):i\in \mathcal {I}\}\) be \(\mathcal {S}\)-measurable?
[Definition: measure] Let \(X\) be a set. Let \(\mathcal {S}\) be a \(\sigma \)-algebra on \(X\). We say that \(\mu \) is a measure on \((X,\mathcal {S})\) if:
We call \((X,\mathcal {S},\mu )\) a measure space.
(Problem 1160) Let \(X\) be a set. Let \(\mu \) satisfy \(\mu (\{x\})=1\) for every \(x\in X\). Can we extend \(\mu \) to a measure on \((X,2^X)\)?
(Problem 1170) Let \(X\) be a set, \(\mathcal {S}\) a \(\sigma \)-algebra on \(X\), and \(w:X\to [0,\infty ]\) a function. Show that \begin {equation*} \mu (E)=\sum _{x\in E} w(x)=\sup \Bigl \{\sum _{x\in D} w(x):D\subseteq E, D\text { finite}\Bigr \} \end {equation*} is a measure.
(Problem 1180) Let \(\mu (E)=0\) if \(E\) is countable and \(\mu (E)=3\) if \(E\) is uncountable. Is \(\mu \) a measure on \((\mathbf {R},2^\mathbf {R})\)?
(Problem 1190) Let \(\mu \) be as in Problem 1180. Is there a \(\sigma \)-algebra \(\mathcal {S}\) on \(\mathbf {R}\) such that \(\mu \big \vert _\mathcal {S}\) is a measure on \(\mathcal {S}\)?
(Problem 1360) Let \((X,\mathcal {S})\) be a measurable space. Let \(\mu \) satisfy
Show that \((X,\mathcal {S},\mu )\) is a measure space.
(Problem 1200) Let \((X,\mathcal {S},\mu )\) be a measure space. Let \(D\), \(E\in \mathcal {S}\) with \(D\subseteq E\). Show that \(\mu (D)\leq \mu (E)\).
(Problem 1210) Show that if \(\mu (D)<\infty \) then \(\mu (E\setminus D)=\mu (E)-\mu (D)\).
(Problem 1220) Give an example of a measure space \((X,\mathcal {S},\mu )\) and sets \(D\), \(E\in \mathcal {S}\) with \(D\subseteq E\), \(\mu (D)=\mu (E)=\infty \), and such that \(\mu (E\setminus D)=0\).
(Problem 1230) Give an example where \(\mu (E\setminus D)=\infty \).
(Problem 1240) Give an example where \(\mu (E\setminus D)=7\).
(Problem 1250) Let \((X,\mathcal {S},\mu )\) be a measure space and let \(\{E_k\}_{k=1}^\infty \subseteq \mathcal {S}\). We do not require that the \(E_k\)s be pairwise-disjoint. Show that \(\mu \Bigl (\bigcup _{k=1}^\infty E_k\Bigr )\leq \sum _{k=1}^\infty \mu (E_k)\).
(Problem 1260) Let \((X,\mathcal {S},\mu )\) be a measure space and let \(\{E_k\}_{k=1}^\infty \subseteq \mathcal {S}\). We require that \(E_k\subseteq E_{k+1}\) for all \(k\). Show that \(\mu \Bigl (\bigcup _{k=1}^\infty E_k\Bigr )=\lim _{k\to \infty } \mu (E_k)\).
(Problem 1270) Let \((X,\mathcal {S},\mu )\) be a measure space and let \(\{E_k\}_{k=1}^\infty \subseteq \mathcal {S}\). We require that \(E_k\supseteq E_{k+1}\) for all \(k\) and that \(\mu (E_k)<\infty \) for at least one \(k\). Show that \(\mu \Bigl (\bigcap _{k=1}^\infty E_k\Bigr )=\lim _{k\to \infty } \mu (E_k)\).
[Exercise 2C.10] Give an example of a measure space \((X,\mathcal {S},\mu )\) and a sequence of sets \(\{E_k\}_{k=1}^\infty \subseteq \mathcal {S}\) such that \(\mu \Bigl (\bigcap _{k=1}^\infty E_k\Bigr )\neq \lim _{k\to \infty } \mu (E_k)\).
(Problem 1280) Let \((X,\mathcal {S},\mu )\) be a measure space and let \(D\), \(E\in \mathcal {S}\). Show that \(\mu (D\cup E)+\mu (D\cap E)=\mu (D)+\mu (E)\).
(Problem 1290) Let \(G\subseteq \mathbf {R}\) be open. Show that \(G\) is the union of countably many pairwise-disjoint open intervals.
(Problem 1300) Let \(X\subseteq \R \) and let \(f:X\to \R \) be strictly increasing. Show that \(f\) is one-to-one.
(Problem 1310) Let \(X\subseteq \R \) and let \(f:X\to \R \) be increasing. Let \(E=\{y\in \R :f^{-1}(y)\) contains more than one element\(\}\). Show that \(E\) is countable.
(Problem 1311) Let \(f:\R \to \R \) be increasing. Let \(E=\{x\in \R :f\) is not continuous at \(x\}\). Show that \(E\) is countable.
(Bonus Problem 1312) Let \(f:\R \to \R \) be increasing. Let \(E=\{y\in \R :y\notin f(\R )\}\). Show that \(E\) is the union of countably many disjoint intervals.
(Problem 1320) Let \(X\subseteq \R \) and let \(f_n:X\to \R \) be increasing for each \(n\). Suppose that there is a function \(f:X\to \R \) such that \(f_n(x)\to f(x)\) for each \(x\in X\). Show that \(f\) is also increasing.
(Problem 1330) Let \(X\), \(Y\) be two metric spaces. Let \(f:X\to Y\) and let \(f_n:X\to Y\) for each \(n\in \mathbf {N}\). Suppose that each \(f_n\) is continuous and that \(f_n\to f\) uniformly on \(X\). Show that \(f\) is continuous.
(Problem 1331) Let \(X\), \(Y\) be two metric spaces. Let \(f:X\to Y\) and let \(f_n:X\to Y\) for each \(n\in \mathbf {N}\). Suppose that each \(f_n\) is uniformly continuous and that \(f_n\to f\) uniformly on \(X\). Show that \(f\) is uniformly continuous.
(Problem 1340) A sequence of functions \(f_n:X\to Y\) is uniformly Cauchy if, for every \(\varepsilon >0\), there is a \(K\in \mathbf {N}\) such that if \(m\), \(n\in \mathbf {N}\) 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. Show that \(f_n\to f\) uniformly for some function \(f:X\to Y\).
(Problem 1350) Let \(\mu (E)=|E|\), where \(|E|\) is the outer measure of Section 2A. Is \(\mu \) a measure on \((\mathbf {R},2^\mathbf {R})\)?
(Problem 1370) Let \(A\), \(Z\subseteq \mathbf {R}\). Suppose that \(|Z|=0\). Show that \(|A\cup Z|=|A|+|Z|\).
(Problem 1371) Let \(A\), \(Z\subseteq \mathbf {R}\). Suppose that \(|Z|=\infty \). Show that \(|A\cup Z|=|A|+|Z|\).
(Problem 1380) Let \(A\subseteq \mathbf {R}\) and \(b<c\), \(b\), \(c\in \mathbf {R}\). Suppose that \(A\cap (b,c)=\emptyset \). Show that \(|A\cup (b,c)|=|A|+|(b,c)|\).
(Problem 1390) Let \(G\subseteq \mathbf {R}\) be open. Suppose \(G=\cup _{k=1}^\infty I_k\) where each \(I_k\) is a (possibly empty or infinite) open interval and the \(I_k\)s are pairwise-disjoint. Show that \(|G|=\sum _{k=1}^\infty \ell (I_k)\).
(Problem 1400) Let \(A\), \(G\subseteq \mathbf {R}\). Suppose that \(A\cap G=\emptyset \) and that \(G\) is open. Show that \(|A\cup G|=|A|+|G|\).
(Problem 1410) Let \(A\), \(F\subseteq \mathbf {R}\). Suppose that \(A\cap F=\emptyset \) and that \(F\) is closed. Show that \(|A\cup F|=|A|+|F|\).
(Problem 1420) Let \(\mathcal {L}=\{D\subseteq \R :{}\)If \(\varepsilon >0\), then there is a closed set \(F\) with \(F\subseteq D\) and with \(|D\setminus F|<\varepsilon \}\). Let \(F\subseteq \R \) be closed. Show that \(F\in \mathcal {L}\).
(Problem 1421) Show that \(D\in \mathcal {L}\) if and only if, for every \(\varepsilon >0\), there is an open set \(G\) with \(D\cup G=\R \) and with \(|G\cap D|<\varepsilon \).
(Problem 1440) Show that if \(\{D_k\}_{k=1}^\infty \subseteq \mathcal {L}\) then \(\bigcap _{k=1}^\infty D_k\in \mathcal {L}\).
(Problem 1440) Let \(F\subseteq \R \) be closed. Suppose that \(|F|<\infty \). Show that \(\R \setminus F\in \mathcal {L}\).
(Problem 1450) Let \(F\subseteq \R \) be closed. Show that \(\R \setminus F\in \mathcal {L}\) even if \(|F|=\infty \).
(Problem 1460) Let \(D\in \mathcal {L}\). Show that \(\R \setminus D\in \mathcal {L}\).
(Problem 1461) Show that if \(D\in \mathcal {L}\) then for all \(\varepsilon >0\), there is an open set \(G\) with \(D\subseteq G\) and with \(|G\setminus D|<\varepsilon \).
(Problem 1462) Show that if for all \(\varepsilon >0\), there is an open set \(G\) with \(D\subseteq G\) and with \(|G\setminus D|<\varepsilon \), then \(D\in \mathcal {L}\).
(Problem 1470) Show that if \(\{D_k\}_{k=1}^\infty \subseteq \mathcal {L}\) then \(\bigcup _{k=1}^\infty D_k\in \mathcal {L}\).
(Problem 1530) Show that the set of all Lebesgue measurable subsets of \(\R \) is a \(\sigma \)-algebra.
(Problem 1480) Show that if \(B\) is a Borel set and \(\varepsilon >0\), then there exists a closed set \(F\) with \(F\subseteq B\) and with \(|B\setminus F|<\varepsilon \).
(Problem 1490) Let \(A\), \(B\subseteq \R \). Suppose that \(B\) is a Borel set and that \(A\cap B=\emptyset \). Show that \(|A\cup B|=|A|+|B|\).
[Exercise 2D.10] Let \(A\), \(B\subseteq \R \). Suppose that \(B\in \mathcal {L}\), where \(\mathcal {L}\) is as in Problem 1420, and that \(A\cap B=\emptyset \). Show that \(|A\cup B|=|A|+|B|\).
(Problem 1500) Show that there exists an \(A\subseteq \R \) that is not a Borel set.
(Problem 1510) Let \(\mathcal {B}\) be the set of all Borel subsets of \(\R \). Let \(\mu (E)=|E|\) for all \(E\in \mathcal {B}\). Show that \((\R ,\mathcal {B},\mu )\) is a measure space.
(Problem 1520) Let \(A\subseteq \R \). Show that the following statements are equivalent:
[Definition: Lebesgue measureable] A set \(A\subseteq \R \) is Lebesgue measurable if there is a Borel set \(B\subseteq A\) such that \(|A\setminus B|=0\).
(Problem 1590) Let \(\mathcal {L}\) be the set of all Lebesgue measureable subsets of \(\R \). Let \(\mu (E)=|E|\) for all \(E\in \mathcal {L}\). Show that \((\R ,\mathcal {L},\mu )\) is a measure space. (We refer to outer measure restricted to \(\mathcal {L}\) as Lebesgue measure.)
(Problem 1600) Show that there exists an \(A\subseteq [0,1]\) that is not Lebesgue measurable.
[Exercise 2D.12] Suppose that \(A\subseteq \R \) is a bounded set. Let \(b\), \(c\in \R \) be such that \(A\subseteq [b,c]\). Suppose that \begin {equation*} |A\cup ([b,c]\setminus A)| = |A|+|[b,c]\setminus A|. \end {equation*} Show that \(A\) is Lebesgue measurable.
[Exercise 2D.13] Suppose that \(A\subseteq \R \). Suppose that \begin {equation*} |(A\cap [-n,n])\cup ([-n,n]\setminus A)| = |A\cap [-n,n]|+|[-n,n]\setminus A| \end {equation*} for all \(n\in \N \). Show that \(A\) is Lebesgue measurable.
(Problem 1610) Suppose that \((\R ,\mathcal {S},\mu )\) is a measure space, where \(\mu (E)=|E|\) for all \(E\in \mathcal {S}\). Suppose further that \(\mathcal {S}\) contains all closed bounded intervals. Show that \(\mathcal {S}\subseteq \mathcal {L}\), where \(\mathcal {L}\) is the set of all Lebesgue measurable sets.
[Definition: Lebesgue measurable] Let \(A\subseteq \R \) and let \(f:A\to \R \). We say that \(f\) is Lebesgue measurable if, whenever \(B\subseteq \R \) is Borel, we have that \(f^{-1}(B)\) is a Lebesgue measurable set (that is, satisfies one of the six equivalent conditions of Problem 1520).
(Problem 1620) Let \(f:\R \to \R \) be increasing and continuous. Let \(\mathcal {S}=\{E\subseteq \R :f(E)\) is Borel\(\}\). Show that \(\mathcal {S}\) contains all intervals.
(Bonus Problem 1621) Show that the previous problem is valid if \(f\) is merely increasing (not necessarily continuous).
(Problem 1630) Show that \(\mathcal {S}\) is a \(\sigma \)-algebra.
(Problem 1640) Show that if \(f:\R \to \R \) is increasing, and if \(B\subseteq \R \) is a Borel set, then \(f(B)\) is also a Borel set.
(Problem 1641) Let \(F_0=F_{1,0}=[0,1]\). If \(n\), \(k\in \N \) and \(F_{k,n-1}=[a_{k,n-1},b_{k,n-1}]\) exists, let \(F_{2k-1,n} = [a_{k,n-1}, \frac {2}{3}a_{k,n-1}+\frac {1}{3}b_{k,n-1}]\), and let \(F_{2k,n} = [ \frac {1}{3}a_{k,n-1}+\frac {2}{3}b_{k,n-1},b_{k,n-1}]\). Show that
Let \(F_n=\bigcup _{k=1}^{2^n} F_{k,n}\).
(Problem 1642) If \(n\in \N \) and \(1<k<2^{n-1}\), let \(G_{k,n} = F_{k,n-1} \setminus (F_{2k-1,n}\cup F_{2k,n})\). Show that
[Definition: The Cantor set] The Cantor set \(C=\bigcap _{k=0}^\infty F_k=[0,1]\setminus \bigcup _{k=1}^\infty G_k\).
(Problem 1650) Show that \(C\) is closed.
(Problem 1660) Show that \(|C|=0\).
(Problem 1670) Let \(I\subseteq \R \) be an interval. Show that if \(I\subseteq C\) then \(I\) contains at most one point.
(Problem 1680) Let \(\Lambda _k(x)=\frac {|F_k\cap (-\infty ,x)|}{|F_k|}\). Show that \(\Lambda _k\) is continuous.
(Problem 1690) Sketch the graphs of \(\Lambda _0\), \(\Lambda _1\), and \(\Lambda _2\).
(Problem 1700) Show that if \(n\geq m\), then \(\Lambda _n(x)=\Lambda _m(x)\) for all \(x\in G_m\).
(Problem 1710) 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 1720) Show that \(\Lambda \) exists and is continuous, increasing, and surjective \(\Lambda :[0,1]\to [0,1]\).
(Problem 1730) Show that \(\Lambda ([0,1]\setminus C)\) is countable.
(Problem 1740) Show that \(\Lambda (C)=[0,1]\).
(Problem 1750) Show that \(C\) is uncountable.
(Problem 1760) Let \(A\subseteq [0,1]\) be a set that is not a Borel set. Let \(E=C\cap \Lambda ^{-1}(A)\). Show that \(E\) is Lebesgue measurable but that \(\Lambda (E)\) is not Lebesgue measurable.
[Definition: Pointwise convergence] Let \(X\) be a set, let \(f:X\to \R \), and let \(\{f_k\}_{k=1}^\infty \) be a sequence of functions from \(X\) to \(R\). We say that the sequence \(\{f_k\}_{k=1}^\infty \) converges to \(f\) pointwise if \(\lim _{k\to \infty } f_k(x)=f(x)\) for each \(x\in X\). That is, if for every \(\varepsilon >0\) and every \(x\in X\) there is a \(N\in \N \) such that \(|f_k(x)-f(x)|<\varepsilon \) for all \(k\geq N\).
[Definition: Uniform convergence] Let \(X\) be a set, let \(f:X\to \R \), and let \(\{f_k\}_{k=1}^\infty \) be a sequence of functions from \(X\) to \(R\). We say that the sequence \(\{f_k\}_{k=1}^\infty \) converges to \(f\) uniformly if for every \(\varepsilon >0\) there is a \(N\in \N \) such that \(|f_k(x)-f(x)|<\varepsilon \) for all \(k\geq N\) and all \(x\in X\).
(Problem 1770) Give an example of a sequence of functions that converges pointwise but not uniformly.
(Problem 1771) Let \(X\) be a metric space and let \(Y\subseteq X\) be a subspace. Show that \(G\subseteq Y\) is open in \(Y\) (relatively open) if and only if there is a \(U\subseteq X\) that is open in \(X\) and satisfies \(G=Y\cap U\).
(Problem 1772) Let \(X\) be a metric space and let \(Y\subseteq X\) be a subspace. Show that \(F\subseteq Y\) is closed in \(Y\) (relatively closed) if and only if there is a \(D\subseteq X\) that is closed in \(X\) and satisfies \(F=Y\cap D\).
(Problem 1773) Let \(X\subseteq \R \) and let \(f:X\to \R \). Show that \(f\) is continuous everywhere on \(X\) if and only if, for every \(U\subseteq \R \) open, the set \(f^{-1}(U)\) is relatively open.
(Problem 1780) Show that if the sequence \(\{f_k\}_{k=1}^\infty \) converges uniformly to \(f\), then the sequence \(\{f_k\}_{k=1}^\infty \) converges pointwise to \(f\).
(Problem 1790) Let \(X\subseteq \R \), let \(x_0\in X\), let \(f:X\to \R \), and let \(\{f_k\}_{k=1}^\infty \) be a sequence of functions from \(X\) to \(R\). Suppose that each \(f_k\) is continuous at \(x_0\) and that the sequence \(\{f_k\}_{k=1}^\infty \) converges to \(f\) uniformly on \(X\). Show that \(f\) is also continuous at \(x_0\).
(Problem 1800) Give an example of a sequence of continuous functions that converge pointwise to a discontinuous function.
(Problem 1810) Let \(F\subseteq \R \) be a closed set. Let \(g:F\to \R \) be continuous. Show that there exists a function \(h:\R \to \R \) such that \(h\) is continuous and such that \(h(x)=g(x)\) for all \(x\in F\).
(Problem 1820) Let \(X\) be a set. Let \(f:X\to \R \) and let \(\{f_k\}_{k=1}^\infty \) be a sequence of functions from \(X\) to \(\R \). Let \(Y=\{x\in X:\lim _{k\to \infty } f_k(x)=f(x)\}\) and let \(A_{n,m,k}=\bigl \{x\in X: |f_k(x)-f(x)|<{\textstyle \frac {1}{n}}\bigr \}\). Write \(Y\) in terms of unions and intersections of the sets \(A_{n,m,k}\).
(Problem 1830) Let \(\{m_n\}_{n=1}^\infty \) be a sequence of natural numbers. Show that \(f_k\) converges uniformly to \(f\) on the set \begin {equation*} \bigcap _{n=1}^\infty \bigcap _{k=m_n}^\infty \bigl \{x\in X: |f_k(x)-f(x)|<{\textstyle \frac {1}{n}}\bigr \} . \end {equation*}
(Problem 1840) Let \((X,\mathcal {S},\mu )\) be a measure space. Let \(f:X\to \R \) be \(\mathcal {S}\)-measurable and let \(\{f_k\}_{k=1}^\infty \) be a sequence of \(\mathcal {S}\)-measurable functions from \(X\) to \(\R \). Suppose that \(f_k\) converges to \(f\) pointwise on \(X\). Let \begin {equation*} A_{n,m}=\bigcap _{k=m}^\infty \bigl \{x\in X: |f_k(x)-f(x)|<{\textstyle \frac {1}{n}}\bigr \}. \end {equation*} Show that \(\lim _{m\to \infty } \mu (A_{n,m})=\mu (X)\).
(Problem 1850) Suppose further that \(\mu (X)<\infty \). Choose some \(\varepsilon >0\). For each \(n\in \N \), let \(m_n\) be such that \(\mu (A_{n,m_n})>\mu (X)-\frac {\varepsilon }{2^n}\). Show that \(\mu (X\setminus \bigcap _{n=1}^\infty A_{n,m_n})<\varepsilon \).
(Problem 1860) [Egorov’s Theorem] Let \((X,\mathcal {S},\mu )\) be a measure space with \(\mu (X)<\infty \). Let \(f:X\to \R \) be \(\mathcal {S}\)-measurable and let \(\{f_k\}_{k=1}^\infty \) be a sequence of \(\mathcal {S}\)-measurable functions from \(X\) to \(\R \). Suppose that \(f_k\) converges to \(f\) pointwise on \(X\). Show that for every \(\varepsilon >0\) there is a set \(E\subseteq X\) with \(\mu (X\setminus E)<\varepsilon \) such that \(f_k\) converges to \(f\) uniformly on \(E\).
[Exercise 2E.5] Give an example of a Borel measurable function \(f:\R \to \R \) and a sequence of Borel measurable functions \(f_k:\R \to \R \) such that \(f_k\to f\) pointwise but such that \(f_k\) does not converge uniformly on any set of infinite measure.
(Problem 1861) What is the best analogue to Egorov’s Theorem available in \(\R \)? In an arbitrary measure space of infinite measure?
[Definition: Simple function] A function is called simple if it takes on only finitely many values.
(Problem 1870) Let \((X,\S )\) be a measurable space and let \(f:X\to \R \) be a simple function. Let \(f(X)=\{c_k:1\leq k\leq n\}\). Let \(E_k=f^{-1}(\{c_k\})\). Show that if \(x\in X\) then \(x\in E_k\) for exactly one value of \(k\) (so \(X=\bigcup _{k=1}^n E_k\) and \(E_j\cap E_k=\emptyset \) if \(j\neq k\)).
(Problem 1871) Show that \(f=\sum _{k=1}^n c_k \,\chi _{E_k}\).
(Problem 1880) Show that \(f\) is \(\S \)-measurable if and only if \(\{E_k:1\leq k\leq n\}\subseteq \S \).
(Problem 1890) Let \(X\) be a set, let \(k\in \N \), and let \(f:X\to \R \). Define \(f_k\) as \begin {equation*} f_k(x)=\sgn (f(x)) \min \left (k,\frac {\lfloor 2^k |f(x)| \rfloor }{2^k}\right ) \end {equation*} where \(\lfloor y\rfloor =\inf \{n\in \Z :n\leq y\}\). Find \(f(X)\) and show that \(f_k\) is simple.
(Problem 1900) Show that \(\{f_k(x)\}_{k=1}^\infty \) is increasing if \(f(x)\geq 0\) and decreasing if \(f(x)\leq 0\).
(Problem 1910) Let \(X_m=\{x\in X:|f(x)|<m\}\). Show that \(f_k\to f\) uniformly on \(X_m\) for each \(m\in \N \).
(Problem 1920) Show that \(\lim _{k\to \infty } f_k(x)=f(x)\) for all \(x\in X\) and that, if \(f\) is bounded, then \(f_k\) converges to \(f\) uniformly on \(X\).
(Problem 1930) Let \(\mathcal {S}\) be a \(\sigma \)-algebra on \(X\). Suppose in addition that \(f\) is \(\mathcal {S}\)-measurable. Show that \(f_k\) is \(\mathcal {S}\)-measurable for each \(k\in \N \).
(Problem 1940) Give an example of a Borel measurable function \(f:\R \to \R \) such that \(f\) is not continuous at any point \(x\in \R \).
(Problem 1950) Let \(B\) be a Lebesgue measurable set. Show that for all \(\varepsilon >0\), there is an open set \(U\) with \(|U|<\varepsilon \) such that \(B\setminus U\) is closed and such that \((\R \setminus B)\setminus U\) is also closed.
(Problem 1960) Show that \(g=\chi _B\big \vert _{\R \setminus U}\) is continuous everywhere on \(\R \setminus U\).
(Problem 1970) Show that if \(s:\R \to \R \) is simple and Lebesgue measurable and \(\varepsilon >0\), then there is an open set \(U\) with \(|U|<\varepsilon \) such that \(s\big \vert _{\R \setminus U}\) is continuous everywhere on \(\R \setminus U\).
(Problem 1980) Let \(f:\R \to \R \) be Lebesgue measurable. Let \(B\subseteq \R \) be Lebesgue measurable with \(|B|<\infty \). For all \(\varepsilon >0\), show that there is an open set \(G\) such that \(|G|<\varepsilon \) and such that \(f\) is bounded on \(B\setminus G\).
(Problem 1990) [Luzin’s theorem] Let \(f:\R \to \R \) be Lebesgue measurable. For all \(\varepsilon >0\), show that there is an open set \(U\) and a continuous function \(h:\R \to \R \) such that \(|U|<\varepsilon \) and such that \(f(x)=h(x)\) for all \(x\notin U\).
(Problem 2000) Need \(f\) be continuous on \(\R \)?
(Problem 2010) Let \(s:\R \to \R \) be Lebesgue measurable and simple. Show that there exists a function \(g:\R \to \R \) that is Borel measurable and such that the set \(\{x\in \R :f(x)\neq g(x)\}\) is Borel and satisfies \(|\{x\in \R :s(x)\neq g(x)\}|=0\).
(Problem 2020) Let \(f:\R \to \R \) be Lebesgue measurable. Show that there exists a function \(g:\R \to \R \) that is Borel measurable and satisfies \(|\{x\in \R :f(x)\neq g(x)\}|=0\).
(Problem 2021) Let \(\{x_n\}_{n=1}^\infty \subseteq [-\infty ,\infty ]\). Suppose that \(x_n\leq x_{n+1}\) for all \(n\). Show that \(\lim _{n\to \infty } x_n\) exists and satisfies \(\lim _{n\to \infty }x_n=\sup _{n\in \N }x_n\).
[Definition: \(\mathcal {S}\)-partition] Let \(\mathcal {S}\) be a \(\sigma \)-algebra on a set \(X\). An \(\mathcal {S}\)-partition of \(X\) is a finite collection of disjoint sets in \(S\) whose union is all of \(X\). (So \(P=\{A_j:1\leq j\leq m\}\), \(m<\infty \), \(A_j\cap A_k=\emptyset \) if \(j\neq k\), and \(X=\cup _{j=1}^m A_j\).)
[Definition: Lower Lebesgue sum] Let \((X,\mathcal {S},\mu )\) be a measure space. Let \(f:X\to [0,\infty ]\) be \(\S \)-measurable. Let \(P=\{A_j:1\leq j\leq m\}\) be an \(\S \)-partition of \(X\). The lower Lebesgue sum \(\mathcal {L}(f,P)\) is \begin {equation*} \mathcal {L}(f,P)=\sum _{j=1}^m \mu (A_j)\inf _{A_j}f. \end {equation*} (If either \(\mu (A_j)=0\) or \(\inf _{A_j} f=0\), then we take \(\mu (A_j)\inf _{A_j}f=0\) even if the other quantity is \(\infty \).)
[Definition: Integral of a nonnegative function] Let \((X,\mathcal {S},\mu )\) be a measure space. Let \(f:X\to [0,\infty ]\) be \(\S \)-measurable. Then \begin {equation*} \int f\,d\mu = \sup \{\mathcal {L}(f,P):P\text { is an $\S $-partition of $X$}\}. \end {equation*}
(Problem 2100) Suppose that \((X,\S ,\mu )\) is a measure space and \(E\in \S \). What is \(\int \chi _E \,d\mu \)?
(Problem 2110) What is \(\int \chi _\Q \,d\lambda \)? What is \(\int \chi _{[0,1]\setminus \Q }\,d\lambda \)?
[Exercise 3A.5] Let \(\{b_k\}_{k=1}^\infty \) be a sequence of nonnegative real numbers. Define the function \(b:\N \to \R \) by \(b(k)=b_k\). Let \(\mu (E)=\#E\) denote the counting measure. Then \(\int b\,d\mu =\sum _{k=1}^\infty b_k\).
(Bonus Problem 2030) Let \(X=[a,b]\) for some \(a\), \(b\in \R \), \(a<b\). Let \(a=x_0<x_1<\dots <x_m=b\). Let \(A_0=\{x_j:0\leq l\leq m\}\) and let \(A_k=(x_{k-1},x_k)\) for \(1\leq k\leq m\). Show that \(P=\{A_j:0\leq j\leq m\}\) is a \(\mathcal {L}\)-partition and a \(\mathcal {B}\)-partition, where \(\mathcal {L}\) and \(\B \) are the \(\sigma \)-algebras of Borel and Lebesgue measurable sets, respectively.
(Bonus Problem 2040) Let \(P\) be the partition in Problem 2030. Let \(f:[a,b]\to \R \) be Lebesgue measurable. How does \(\mathcal {L}(f,P)\) relate to the upper and lower Darboux sums \(U(f,A_0)\) and \(L(f,A_0)\) of \(f\) over \(A_0=\{x_0,\dots ,x_m\}\)?
(Bonus Problem 2050) Suppose that \(X=[a,b]\), \(\S \) denotes the Borel (or Lebesgue) sets, \(\lambda \) denotes Lebesgue measure, and \(f:X\to [0,\infty )\) is bounded and Borel measurable. How does \(\int f\,d\lambda \) compare to the lower Riemann integral \(L(f,[a,b])\)?
(Bonus Problem 2060) Let \((X,\S ,\mu )\) be a measurable space and \(f:X\to [0,\infty )\) be bounded. How would you define an upper Lebesgue sum \(\mathcal {U}(f,P)\)? How would you define an integral in terms of the upper Lebesgue sum?
(Bonus Problem 2061) Let \(P\), \(Q\) be two partitions of \(X\). What can you say about \(\mathcal {L}(f,P)\) and \(\mathcal {U}(f,Q)\)?
[Exercise 3B.4a] If \(\mu (X)<\infty \) and \(f:X\to [0,\infty )\) is a bounded \(\S \)-measurable function, then the “upper” and “lower” Lebesgue integrals are equal.
[Exercise 3B.4b]
[Exercise 3B.4c]
(Bonus Problem 2080) Why did we use the “lower” Lebesgue integral (instead of the “upper” Lebesgue integral) as the definition of Lebesgue integral?
(Bonus Problem 2090) Suppose that \(X=[a,b]\), \(\S \) denotes the Borel (or Lebesgue) subsets of \(X\), \(\lambda \) denotes Lebesgue measure, and \(f:X\to [0,\infty )\) is bounded and Borel measurable. How does the (upper) Lebesgue integral compare to the upper Riemann integral \(U(f,[a,b])\)?
(Bonus Problem 2091) Suppose that \(X=[a,b]\), \(\S \) denotes the Borel (or Lebesgue) subsets of \(X\), \(\lambda \) denotes Lebesgue measure, and \(f:X\to [0,\infty )\) is bounded. Show that if \(f\) is Riemann integrable, then \(f\) is Lebesgue measurable, and moreover the Riemann and Lebesgue integrals of \(f\) coincide.
(Bonus Problem 2092) Give an example of a function that is Lebesgue integrable (measurable) but not Riemann integrable.
(Problem 2120) Let \((X,\S ,\mu )\) be a measure space. Let \(f\) be an \(\S \)-measurable simple function, so \(f=\sum _{k=1}^m c_k\chi _{E_k}\) for some partition \(\{E_k\}\) and distinct values \(c_k\). What is \(\int f \,d\mu \) in terms of \(c_k\) and \(E_k\)?
(Problem 2121) Suppose that \(f=\sum _{k=1}^m c_k\chi _{E_k}\), where the \(E_k\)s are pairwise-disjoint, but possibly empty or not having union \(X\), and where the numbers \(c_k\) need not be distinct. What is \(\int f \,d\mu \)?
(Bonus Problem 2122) What is \(\int f \,d\mu \) if we relax the requirement that the \(E_k\)s be pairwise disjoint?
(Problem 2130) Let \((X,\S ,\mu )\) be a measure space. Let \(f:X\to [0,\infty ]\) be \(\S \)-measurable. Show that \begin {equation*} \int f\,d\mu = \sup \biggl \{\int s\,d\mu : s\text { is simple and } 0\leq s(x)\leq f(x)\text { for all }x\in X\biggr \}. \end {equation*}
(Problem 2140) Let \((X,\S ,\mu )\) be a measure space. Let \(f\), \(g:X\to [0,\infty ]\) be \(\S \)-measurable. Suppose that \(f(x)\leq g(x)\) for all \(x\in X\). Show that \(\int f\,d\mu \leq \int g\,d\mu \).
(Problem 2150) Let \((X,\S ,\mu )\) be a measure space. Let \(f_k:X\to [0,\infty ]\) be \(\S \)-measurable. Suppose that \(f_k(x)\leq f_{k+1}(x)\) for all \(x\in X\) and all \(k\in \N \). Show that \(\lim _{k\to \infty } \int f_k \,d\mu \) exists.
(Problem 2160) Let \(f(x)=\lim _{k\to \infty }f_k(x)\). By Problem 1120, \(f\) is \(\S \)-measurable. Show that \(\lim _{k\to \infty } \int f_k \,d\mu \leq \int f\,d\mu \).
(Problem 2170) Let \(s\leq f\) be a \(\S \)-measurable simple function. Let \(0<t<1\). Let \begin {equation*} s_k(x)=\begin {cases}t\,s(x),&\text {if }t\,s(x)\leq f_k(x),\\ 0, & \text {otherwise}.\end {cases} \end {equation*} Show that \(s_k\) is simple and \(\S \)-measurable.
(Problem 2171) What can you say about \(\int s_k \,d\mu \) and \(\int f_k\,d\mu \)?
(Problem 2180) What is \(\lim _{k\to \infty }\int s_k\,d\mu \)?
(Problem 2190) Show that \(\lim _{k\to \infty } \int f_k \,d\mu =\int f\,d\mu \).
[Exercise 3A.8] There exists a sequence of simple Borel measurable functions from \(\R \) to \([0,\infty )\) such that \(\lim _{k\to \infty } f_k(x)=0\) for all \(x\in \R \) but \(\lim _{k\to \infty } \int f_k \,d\lambda =1\).
(Problem 2240) Let \((X,\S ,\mu )\) be a measure space. Let \(f\) be a nonnegative \(\S \)-measurable function. Let \(c\geq 0\). Show that \(\int cf \,d\mu = c\int f\,d\mu \).
(Problem 2210) Let \((X,\S ,\mu )\) be a measure space. Let \(f\), \(g\) be two nonnegative \(\S \)-measurable simple functions. Show that \(f+g\) is also simple.
(Problem 2220) Show that \(\int f+g \,d\mu = \int f\,d\mu + \int g \,d\mu \) if \(f\) and \(g\) are nonnegative and simple.
(Problem 2230) Let \((X,\S ,\mu )\) be a measure space. Let \(f\), \(g\) be two nonnegative \(\S \)-measurable functions. By Problem 1090, \(f+g\) is also \(\S \)-measurable. Show that \(\int f+g \,d\mu = \int f\,d\mu + \int g \,d\mu \).
[Definition: Positive and negative parts] Let \(f:X\to [-\infty ,\infty ]\) be a function. We define \begin {equation*} f^+(x)=\begin {cases}f(x) &\text {if }f(x)>0,\\0&\text {if }f(x)\leq 0,\end {cases}\quad f^-(x)=\begin {cases}-f(x) &\text {if }f(x)<0,\\0&\text {if }f(x)\geq 0.\end {cases} \end {equation*}
(Problem 2250) Show that \(f(x)=f^+(x)-f^-(x)\) and that \(f^\pm :X\to [0,\infty ]\).
(Problem 2260) Let \((X,\S )\) be a measurable space and \(f:X\to [-\infty ,\infty ]\). Suppose \(f\) is \(\S \)-measurable. Show that \(f^+\) and \(f^-\) are \(\S \)-measurable.
[Definition: Integral of a real-valued function] Let \((X,\S ,\mu )\) be a measure space. Let \(f:X\to [-\infty ,\infty ]\) be \(\S \)-measurable. If \(\int f^+\,d\mu <\infty \) or \(\int f^- \,d\mu <\infty \) (or both), we define \begin {equation*} \int f\,d\mu = \int f^+\,d\mu -\int f^-\,d\mu . \end {equation*}
(Problem 2290) What are the two ways that \(\int f\,d\mu \) can fail to exist?
(Problem 2270) Show that the definition of the integral of a nonnegative function given above coincides with this new definition in the case where \(f\) is nonnegative.
(Problem 2280) Show that \(\int |f|\,d\mu <\infty \) if and only if \(\int f^+\,d\mu <\infty \) and \(\int f^-\,d\mu <\infty \).
(Problem 2330) Suppose \((X,\S ,\mu )\) is a measure space, \(f:X\to [-\infty ,\infty ]\), and \(\int f\,d\mu \) exists. Show that \(\left |\int f\,d\mu \right |\leq \int |f|\,d\mu \).
(Problem 2300) Suppose \((X,\S ,\mu )\) is a measure space, \(f:X\to [-\infty ,\infty ]\), and \(\int f\,d\mu \) exists. Let \(c\in \R \). Show that \(\int cf\,d\mu = c\int f\,d\mu \).
(Problem 2310) Suppose \((X,\S ,\mu )\) is a measure space, \(f\), \(g:X\to [-\infty ,\infty ]\), and \(\int |f|\,d\mu <\infty \), \(\int |g|\,d\mu <\infty \). Show that \(\int f+g \,d\mu = \int f\,d\mu + \int g \,d\mu \).
(Problem 2320) Suppose \((X,\S ,\mu )\) is a measure space, \(f\), \(g:X\to [-\infty ,\infty ]\), \(f(x)\leq g(x)\) for all \(x\in X\), and \(\int f\,d\mu \), \(\int g\,d\mu \) exist. Show that \(\int f\,d\mu \leq \int g\,d\mu \).
[Definition: Integration on a subset] Let \((X,\S ,\mu )\) be a measure space. Let \(E\in \S \). If \(f:X\to [-\infty ,infty]\) is a \(\S \)-measurable function, then we let \begin {equation*} \int _E f\,d\mu = \int \chi _E f\,d\mu \end {equation*} provided the right-hand side exists.
(Problem 2350) Let \(A\), \(B\in \S \) be disjoint. Show that \(\int _A f\,d\mu + \int _B f\,d\mu = \int _{A\cup B} f\,d\mu \).
(Problem 2340) Show that \(\left |\int _E f\,d\mu \right | \leq \mu (E)\sup _E |f|\).
(Problem 2360) [The Bounded Convergence Theorem] Let \((X,\S ,\mu )\) be a measure space. Let \(\{f_k\}_{k=1}^\infty \) be a sequence of \(\S \)-measurable functions from \(X\) to \(\R \) that converges pointwise on \(X\) to a function \(f:X\to \R \). Suppose that \(\mu (X)<\infty \) and that \(\sup _{x\in X}\sup _{k\in \N } |f_k(x)|<\infty \). Show that \(\lim _{k\to \infty } \int f_k \,d\mu = \int f\,d\mu \).
(Problem 2380) Suppose that \(\mu (E)=0\). Show that \(\int _E f\,d\mu =0\).
(Problem 2370) Let \((X,\S ,\mu )\) be a measure space. Let \(f\), \(g:X\to [-\infty ,\infty ]\) be two \(\S \)-measurable functions. Suppose that \(\mu \{x\in X:f(x)\neq g(x)\}=0\). Show that \(\int f\,d\mu \) exists if and only if \(\int g\,d\mu \) exists, and that \(\int f\,d\mu =\int g\,d\mu \) if they exist.
[Definition: Almost everywhere] Let \((X,\S ,\mu )\) be a measure space. If \(E\in \S \), then \(E\) contains \(\mu \)-almost every element in \(X\) if \(\mu (X\setminus E)=0\). If \(\mu \) is clear from context, we say \(E\) contains almost every element in \(X\).
(Problem 2390) Show that the Bounded Convergence Theorem is still true if we relax the requirement that the \(f_k\)s are uniformly bounded to the requirement that there is some \(c\in \R \) such that \(|f_k(x)|\leq c\) for \(\mu \)-almost every \(x\in X\).
(Problem 2400) Let \((X,\S ,\mu )\) be a measure space. Let \(g:X\to [0,\infty ]\) be \(\S \)-measurable. Suppose that \(g\) is bounded. Show that for every \(\varepsilon >0\) there is a \(\delta >0\) such that, if \(B\in \S \) and \(\mu (B)<\delta \), then \(\int _B g\,d\mu <\varepsilon \).
(Problem 2410) Show that the preceding problem is true provided \(\int g\,d\mu <\infty \) even if \(g\) is not bounded.
[Exercise 3A.1] Let \((X,\S ,\mu )\) be a measure space. Let \(g:X\to [0,\infty ]\) be \(\S \)-measurable. Suppose that \(\int g\,d\mu <\infty \). Show that if \(E\in \S \) and \(\mu (E)=\infty \) then \(\inf _E g =0\).
(Problem 2420) Show that for every \(\varepsilon >0\) there is an \(E\in \S \) with \(\mu (E)<\infty \) and \(\int _{X\setminus E} g\,d\mu <\varepsilon \).
(Problem 2430) [The Dominated Convergence Theorem] Let \((X,\S ,\mu )\) be a measure space, and let \(f\), \(f_k:X\to [-\infty ,\infty ]\), \(g:X\to [0,\infty ]\) be \(\S \)-measurable. Suppose that
Show that \(\lim _{k\to \infty } \int f_k\,d\mu =\int f\,d\mu \).
(Problem 2440) [The Dominated Convergence Theorem] Show that the previous problem is true even if \(\mu (X)=\infty \).
[Definition: The Lebesgue space] Let \((X,\S ,\mu )\) be a measure space. Let \(f:X\to [-\infty ,\infty ]\) be \(\S \)-measurable. Then the \(L^1\)-norm of \(f\) is defined by \begin {equation*} \|f\|_1 = \int |f|\,d\mu . \end {equation*} The Lebesgue space \(L^1(\mu )\) is defined by \begin {equation*} L^1(\mu )=\{f:f\text { is an $\S $-measurable function $f:X\to \R $ and $\|f\|_1<\infty $}\}. \end {equation*}
(Problem 2450) Let \(f\), \(g\in L^1(\mu )\). Show that
(Problem 2451) Is \(\|\,\cdot \,\|_1\) a norm on the vector space \(L^1(\mu )\)?
(Problem 2460) Let \(\mu \) denote the counting measure on \(\N \). What is \(L^1(\mu )\)? (This space is often called \(\ell ^1\).)
(Problem 2470) Let \(f\in L^1(\mu )\) and let \(\varepsilon >0\). Show that there is a simple function \(s\in L^1(\mu )\) that satisfies \(\|f-s\|_1<\varepsilon \).
[Definition: Lebesgue space on the real numbers] \(L^1(\R )=L^1(\lambda )\) where \(\lambda \) denotes Lebesgue measure. (The underlying \(\sigma \)-algebra is the Lebesgue or Borel measurable sets.)
[Definition: Step function] We say that \(g:\R \to \R \) is a step function if \(g=\sum _{j=1}^m a_j\chi _{I_j}\), where each \(I_j\) is an interval and each \(a_j\in \R \).
(Problem 2480) Let \(g\) be a step function. Show that we may require the intervals \(I_j\) to be pairwise disjoint.
(Problem 2500) Let \(f\in L^1(\R )\) and let \(\varepsilon >0\). Show that there is a step function \(s\in L^1(\R )\) that satisfies \(\|f-s\|_1<\varepsilon \).
(Problem 2510) Let \(f\in L^1(\R )\) and let \(\varepsilon >0\). Show that there is a continuous function \(g\in L^1(\R )\) that satisfies \(\|f-g\|_1<\varepsilon \).
(Bonus Problem 2511) Can we require that \(g\) have compact support, that is, that \(g(x)=0\) outside of some bounded set?
(Bonus Problem 2512) Can we require that \(g\) be continuously differentiable? Twice differentiable? Smooth (differentiable to order \(m\) for all \(m\in \N \))?
(Problem 2520) (Markov’s inequality) Let \((X,\S ,\mu )\) be a measure space, let \(h\in L^1(\mu )\), and let \(c\in (0,\infty )\). Show that \begin {equation*} \mu \{x\in X:|h(x)|\geq c\}\leq \frac {1}{c}\|h\|_1. \end {equation*}
[Definition: \(3I\)] Let \(I\) be a bounded nonempty open interval in \(\R \). Then \(3I\) is the open interval with the same center as \(I\) and three times its length.
(Problem 2530) Show that if \(I\), \(J\) are bounded nonempty non-disjoint open intervals, and \(\ell (I)\geq \ell (J)\), then \(J\subseteq 3I\).
(Bonus Problem 2540) Let \(I_1=(0,10)\), \(I_2=(9,15)\), \(I_3=(14,22)\), \(I_4=(21,31)\). What subsets of \(\{I_1,I_2,I_3,I_4\}\) are pairwise disjoint?
(Bonus Problem 2550) Find \(I_1\cup I_2\cup I_3\cup I_4\).
(Bonus Problem 2560) Find \(\bigcup _{I\in \mathcal {J}}3I\) for each of the sets \(\mathcal {J}\) you found in Problem 2540.
(Problem 2570) [The Vitali covering lemma] Let \(\{I_k\}_{k=1}^n\) be a list of finitely many bounded nonempty open intervals in \(\R \). Show that there exists a sublist \(\{I_{k_j}\}_{j=1}^m\) such that the \(I_{k_j}\)s are pairwise disjoint and \(\bigcup _{k=1}^n I_k\subseteq \bigcup _{j=1}^m 3I_{k_j}\).
[Definition: Hardy-Littlewood maximal function] Let \(h:\R \to \R \) be measurable. Then \begin {equation*} h^*(b)=\sup _{t>0} \frac {1}{2t}\int _{b-t}^{b+t} |h|. \end {equation*}
(Problem 2571) Show that if \(h\) is bounded and \(b\in \R \) then \(h^*(b)\leq \sup _{\R }|h|\).
(Bonus Problem 2580) Let \(h=\chi _{[0,1]}\). Find \(h^*\).
[Exercise 4A.9] Let \(h:\R \to \R \) be measurable and let \(c>0\). Then \(\{b\in \R :h^*(b)>c\}\) is open.
(Problem 2590) Suppose \(h\in L^1(\R )\) and \(c\in (0,\infty )\). Show that \begin {equation*} |\{b\in \R :h^*(b)>c\}|<\frac {3}{c}\|h\|_1. \end {equation*}
[Definition: Derivative] Let \(I\subseteq \R \) be an open interval, \(b\in I\) and \(g:I\to \R \). Then \(g'(b)=\lim _{t\to 0} \frac {g(b+t)-g(b)}{t}\) if the limit exists, in which case \(g\) is differentiable at \(b\).
(Problem 2600) Let \(f:\R \to \R \) be continuous. Show that if \(b\in \R \) then \begin {equation*} \lim _{t\to 0^+} \frac {1}{2t}\int _{b-t}^{b+t} |f-f(b)|=0. \end {equation*}
(Problem 2620) Let \(f\in L^1(\R )\) be continuous. Define \(g(x)=\int _{-\infty }^x f\). Let \(b\in \R \). Show that \(g\) is differentiable at \(b\) and that \(g'(b)=f(b)\).
(Problem 2610) Let \(f\in L^1(\R )\). Show that \begin {equation*} \lim _{t\to 0^+} \frac {1}{2t}\int _{b-t}^{b+t} |f-f(b)|=0 \end {equation*} for almost every \(b\in \R \).
(Problem 2660) Let \(f\in L^1(\R )\). Show that \(f(b)=\lim _{t\to 0^+} \frac {1}{2t} \int _{b-t}^{b+t} f\) for almost every \(b\in \R \).
(Problem 2630) Let \(f\in L^1(\R )\). Define \(g(x)=\int _{-\infty }^x f\). Let \(b\in \R \). Show that \(g\) is differentiable at \(b\) with \(g'(b)=f(b)\) for almost every \(b\in \R \).
(Problem 2640) Let \(E\subseteq [0,1]\). Suppose that \(E\) has the property that \(|E\cap [0,b]|=\frac {b}{2}\) for all \(b\in [0,1]\). Show that \(E\) is not Lebesgue measurable.
(Bonus Problem 2650) Can you show that such a set \(E\) exists?
[Definition: Density] Let \(E\subseteq \R \) and \(b\in \R \). The density of \(E\) at \(b\) is \(\lim _{t\to 0^+}\frac {|E\cap (b-t,b+t)|}{2t}\) provided the limit exists.
(Problem 2670) Let \(E\subseteq \R \) be a Lebesgue measurable set with \(|E|<\infty \). Show that the density of \(E\) is \(1\) at almost every \(b\in E\) and that the density of \(E\) is \(0\) at almost every \(b\notin E\).
(Problem 2680) Show that the above result is still true even if \(|E|=\infty \).
(Problem 2690) Let \(G\subseteq \R \) be open and nonempty. Show that there exist two closed sets \(F\), \(\widehat F\subseteq G\setminus \Q \) with \(F\cap \widehat F=\emptyset \) and \(|F|>0\), \(|\widehat F|>0\).
(Problem 2700) Let \(S\) be the set of all nonempty bounded open intervals in \(\R \) with rational endpoints. Is \(S\) countable or uncountable?
(Problem 2710) If \(G\subseteq \R \) is open, is there an \(I\in S\) with \(I\subseteq G\)?
(Problem 2720) Show that there exists a Borel set \(E\) such that \(0<|E\cap I|<|I|\) for all nonempty bounded open intervals \(I\).
(Problem 2730) Why don’t Problems 2680 and 2720 contradict each other?
[Definition: Cartesian product] Let \(A\) and \(B\) be sets. The Cartesian product \(A\times B\) of \(A\) and \(B\) is the set of all ordered pairs \((a,b)\) with \(a\in A\) and \(b\in B\).
[Definition: Rectangle] If \(X\) and \(Y\) are sets, then \(R\subseteq X\times Y\) is called a rectangle if there exist sets \(A\subseteq X\), \(B\subseteq Y\) with \(R=A\times B\).
[Definition: Measurable rectangle] Let \((X,\S )\) and \((Y,\mathcal {T})\) be measurable spaces. A set \(R\subseteq X\times Y\) is called a measurable rectangle if there exist sets \(A\in \S \), \(B\in \mathcal {T}\) with \(R=A\times B\).
[Definition: Product \(\sigma \)-algebra] Let \((X,\S )\) and \((Y,\mathcal {T})\) be measurable spaces. Then \(\S \otimes \mathcal {T}\) is the smallest \(\sigma \)-algebra on \(X\times Y\) that contains all measurable rectangles in \(X\times Y\).
(Problem 2740) How is \(\S \otimes \mathcal {T}\) different from \(\S \times \mathcal {T}\)?
[Definition: Cross section of a set] Let \(X\) and \(Y\) be sets and let \(E\subseteq X\times Y\). If \(a\in X\) or \(b\in Y\), then the cross sections \([E]_a\) and \([E]^b\) are defined by \begin {equation*} [E]_a=\{y\in Y:(a,y)\in E\},\quad [E]^b=\{x\in X:(x,b)\in E\}. \end {equation*}
(Problem 2750) Let \(X=Y=\R \) and let \(E=\{(x,y):x^2+y^2<25\}\). Draw \([E]_3\) and \([E]^4\) in \(\R ^2=\R \times \R \).
(Problem 2760) Suppose \(a\in X\), \(A\subseteq X\) and \(b\in Y\), \( B\subseteq Y\). What are \([A\times B]_a\) and \([A\times B]^b\)?
(Problem 2770) Let \((X,\S )\) and \((Y,\mathcal {T})\) be measurable spaces. Let \begin {equation*} \mathcal {E}=\{E\subseteq X\times Y:[E]_a\in \mathcal {T},\> [E]^b\in \mathcal {S}\text { for all }a\in X,\>b\in Y\}. \end {equation*} Show that \(\mathcal {E}\) contains all measurable rectangles in \(X\times Y\).
(Problem 2780) Show that \(\mathcal {E}\) is a \(\sigma \)-algebra.
(Problem 2790) What can you conclude about \(\mathcal {E}\) and \(\S \otimes \T \)?
(Problem 2791) Is \(\mathcal {E}\subseteq \S \otimes \T \)?
[Definition: Cross sections of functions] Let \(f:X\times Y\to \R \) be a function. If \(a\in X\) and \(b\in Y\), we let \begin {equation*} [f]_a(y)=f(a,y),\quad [f]^b(x)=f(x,b) \end {equation*} for all \(y\in Y\), \(x\in X\).
(Problem 2800) Let \(A\subseteq X\), \(B\subseteq Y\), and \(f=\chi _{A\times B}\). If \(a\in X\), what is \([f]_a\)? If \(b\in B\), what is \([f]^b\)?
(Problem 2801) If \(a\in X\) and \(E\subseteq X\times Y\), what can you say about \([\chi _E]_a\) and \(\chi _{[E]_a}\)?
(Problem 2810) Let \((X,\S )\) and \((Y,\mathcal {T})\) be measurable spaces. Let \(a\in X\) and \(b\in Y\). Let \(f:X\times Y\to \R \) be a \(\S \otimes \T \)-measurable function. Show that \([f]_a\) is \(\T \)-measurable and \([f]^b\) is \(\S \)-measurable.
[Definition: Algebra] Let \(W\) be a set and let \(\mathcal {A}\) be a set of subsets of \(W\). We say that \(\mathcal {A}\) is an algebra on \(W\) if:
(Problem 2840) Show that all algebras are closed under finite intersections.
(Problem 2850) Let \((X,\S )\) and \((Y,\mathcal {T})\) be measurable spaces. Let \(\mathcal {A}\) be the set of all finite unions of measurable rectangles in \(\S \otimes \T \). Show that \(\mathcal {A}\) is an algebra on \(X\times Y\).
(Problem 2851) Let \(A\times B\) and \(C\times D\) be two measurable rectangles in \(X\times Y\). Show that \((A\times B)\cup (C\times D)=(A\times B)\cup H\cup J\), where \(H\) and \(J\) are two measurable rectangles that satisfy \((A\times B)\cap H=(A\times B)\cap J=H\cap J=\emptyset \).
(Problem 2860) Let \(E\) be a finite union of measurable rectangles. Show that \(E\) is a finite union of disjoint measurable rectangles.
[Definition: Monotone class] Let \(W\) be a set and let \(\mathcal {M}\) be a set of subsets of \(W\). We say that \(\mathcal {M}\) is a monotone class on \(W\) if:
(Problem 2870) Show that the set of all intervals in \(\R \) is a monotone class.
(Problem 2880) Show that the set of all intervals in \(\R \) is not an algebra.
(Problem 2890) Let \(\mathcal {C}\) be a collection of monotone classes on a set \(W\). Show that \(\bigcap _{\mathcal {M}\in \mathcal {C}} \mathcal {M}\) is a monotone class on \(W\).
[Definition: Smallest monotone class] Let \(\mathcal {B}\) be a set of subsets of a set \(W\). Let \(\mathcal {C}=\{\mathcal {M}:\mathcal {B}\subseteq \mathcal {M}\subseteq 2^W,\>\mathcal {M}\) is a monotone class\(\}\). We call \(\bigcap _{\mathcal {M}\in \mathcal {C}} \mathcal {M}\) the smallest monotone class containing \(\mathcal {B}\).
(Problem 2900) Let \(\mathcal {B}\) be a set of subsets of a set \(W\). Show that the smallest \(\sigma \)-algebra containing \(\mathcal {B}\) contains the smallest monotone class containing \(\mathcal {B}\).
(Problem 2910) Let \(\mathcal {A}\) be an algebra on a set \(W\). Let \(\mathcal {M}\) be the smallest monotone class containing \(\mathcal {A}\). Show that \(\mathcal {M}\) is a \(\sigma \)-algebra.
(Problem 2911) Show that \(\M \) is the smallest \(\sigma \)-algebra containing \(\A \).
[Definition: Finite measure] A measure \(\mu \) on a measurable space \((X,\S )\) is finite if \(\mu (X)<\infty \).
[Definition: \(\sigma \)-finite] A measure \(\mu \) on a measurable space \((X,\S )\) is \(\sigma \)-finite if there exists a sequence \(\{X_k\}_{k=1}^\infty \subseteq \S \) with \(X=\bigcup _{k=1}^\infty X_k\) and \(\mu (X_k)<\infty \) for each \(k\in \N \).
(Problem 2912) Show that we may also require \(X_k\subseteq X_{k+1}\) for all \(k\in \N \).
(Problem 2920) Give an example of a finite measure.
(Problem 2930) Show that Lebesgue measure on \(\R \) is \(\sigma \)-finite but not finite.
(Problem 2940) Show that the counting measure on \(\N \) is \(\sigma \)-finite but not finite.
(Problem 2950) Show that the counting measure on \(\R \) is not \(\sigma \)-finite.
(Problem 2960) Let \((X,\S ,\mu )\) and \((Y,\T ,\nu )\) be measure spaces. Suppose \(\mu (X)<\infty \). If \(E\in \S \otimes \T \), show that \(y\mapsto \mu ([E]^y)\) is a \(\T \)-measurable function on \(Y\).
(Problem 2970) Show that the result is still true if \((X,\S ,\mu )\) is a \(\sigma \)-finite measure space.
[Definition: Integral notation] If \((X,\S ,\mu )\) is a measure space and \(g:X\to [-\infty ,\infty ]\) is a \(\S \)-measurable function then \begin {equation*} \int _X g(x)\,d\mu (x)=\int g\,d\mu \end {equation*} where \(d\mu (x)\) indicates that variables other than \(x\) should be treated as constants.
[Definition: Product of measures] Let \((X,\S ,\mu )\) and \((Y,\T ,\nu )\) be \(\sigma \)-finite measure spaces. Let \(E\in \S \otimes \T \). We define \begin {align*} (\mu \times \nu )(E) &=\int _Y \mu ([E]^y)\,d\nu (y) . \end {align*}
(Problem 2980) Let \(A\in \S \) and \(B\in \T \). Show that \((\mu \times \nu )(A\times B)=\mu (A)\cdot \nu (B)\).
(Problem 2981) Show that if \(E\in \S \otimes \T \) then \((\mu \times \nu )(E)\) is well defined and nonnegative.
(Problem 2990) Show that \(\mu \times \nu \) is a measure on \((X\times Y,\S \otimes \T )\).
(Problem 2991) Let \(E_1\subseteq E_2\subseteq \dots \) be an increasing sequence of sets in \(\S \otimes \T \). Show that \[\int _Y \mu ([\bigcup _{k=1}^\infty E_k]^y)\,d\nu (y) =\lim _{k\to \infty }\int _Y \mu ([E_k]^y)\,d\nu (y).\]
(Problem 2992) Let \(E_1\supseteq E_2\supseteq \dots \) be a decreasing sequence of sets in \(\S \otimes \T \). If \(\mu (X)<\infty \), show that \[\int _Y \mu ([\bigcap _{k=1}^\infty E_k]^y)\,d\nu (y) =\lim _{k\to \infty }\int _Y \mu ([E_k]^y)\,d\nu (y).\]
(Problem 3000) Suppose \(\mu (X)<\infty \) and \(\mu (Y)<\infty \). Show that \begin {equation*} \int _Y \mu ([E]^y)\,d\nu (y) = \int _X \nu ([E]_x)\,d\mu (x) \end {equation*} for all \(E\in \S \otimes \T \).
(Problem 3010) Show that \begin {equation*} \int _Y \mu ([E]^y)\,d\nu (y) = \int _X \nu ([E]_x)\,d\mu (x) \end {equation*} for all \(E\in \S \otimes \T \) if \(X\) and \(Y\) are \(\sigma \)-finite (and not necessarily finite).
(Problem 3020) Let \((X,\S ,\mu )\) and \((Y,\T ,\nu )\) be \(\sigma \)-finite measure spaces. Let \(f:X\times Y\to [0,\infty ]\) be a \(\S \otimes \T \)-measurable function. Let \(g(x)=\int _Y f(x,y)\,d\nu (y)=\int _Y [f]_x(y)\,d\nu (y)\). Show that \(g:X\to \R \) is \(\S \)-measurable.
[Definition: Iterated integral] Let \((X,\S ,\mu )\) and \((Y,\T ,\nu )\) be measure spaces. Let \(f:X\times Y\to [-\infty ,\infty ]\) be a \(\S \otimes \T \)-measurable function. We define \begin {align*} \int _X \int _Y f(x,y)\,d\nu (y)\,d\mu (x) &=\int _X \biggl (\int _Y f(x,y)\,d\nu (y)\biggr )d\mu (x) \\&=\int _X \biggl (\int _Y [f]_x\,d\nu (y)\biggr )d\mu . \end {align*}
(Problem 3030) [Tonelli’s Theorem] Let \((X,\S ,\mu )\) and \((Y,\T ,\nu )\) be \(\sigma \)-finite measure spaces. Let \(f:X\times Y\to [0,\infty ]\) be a \(\S \otimes \T \)-measurable function. Show that \[\int _{X\times Y} f\,d(\mu \times \nu )=\int _X\int _Y f(x,y)\,d\nu (y)\,d\mu (x) =\int _Y\int _X f(x,y)\,d\mu (x)\,d\nu (y).\]
[Exercise 5B.1] Tonelli’s theorem can fail for measurable functions \(f:X\times Y\to \R \).
(Problem 3040) Let \(\mathcal {B}\) be the \(\sigma \)-algebra of Borel subsets of \([0,1]\). Let \(\mu \) denote the counting measure. Is \(([0,1],\mathcal {B},\mu )\) \(\sigma \)-finite?
(Problem 3050) Let \(\lambda \) denote Lebesgue measure on \(\mathcal {B}\). Let \(D=\{(x,x):x\in [0,1]\}\). Find \[\int _{[0,1]}\int _{[0,1]} \chi _D(x,y)\,d\mu (y)\,d\lambda (x).\]
(Problem 3060) Find \[\int _{[0,1]}\int _{[0,1]} \chi _D(x,y)\,d\lambda (x)\,d\mu (y).\]
(Problem 3070) Show that if \(x_{j,k}\geq 0\) for all \(j\), \(k\in \N \), then \(\sum _{j=1}^\infty \sum _{k=1}^\infty x_{j,k} = \sum _{k=1}^\infty \sum _{j=1}^\infty x_{j,k} \).
(Problem 3080) Let \((X,\S ,\mu )\) and \((Y,\T ,\nu )\) be \(\sigma \)-finite measure spaces. Let \(f:X\times Y\to [-\infty ,\infty ]\) be a \(\S \otimes \T \)-measurable function. Suppose that \(\int _{X\times Y} |f| d(\mu \times \nu )<\infty \). Show that \(\int _X|f(x,y)|\,d\mu (x)<\infty \) for \(\nu \)-almost every \(y\in Y\) and that \(\int _Y|f(x,y)|\,d\nu (y)<\infty \) for \(\mu \)-almost every \(x\in X\).
(Problem 3090) What can you say about the set \(\{x\in X:\int _Y f(x,y)\,d\nu (y)\) does not exist\(\}\)?
(Problem 3100) Show that \(x\mapsto \int _Y f(x,y)\,d\nu (y)\) is a \(\S \)-measurable function and that \(y\mapsto \int _X f(x,y)\,d\mu (x)\) is a \(\T \)-measurable function (up to a set of measure zero).
(Problem 3110) [Fubini’s theorem] Show that if \(\int _{X\times Y} |f| d(\mu \times \nu )<\infty \) then \[\int _{X\times Y} f\,d(\mu \times \nu )=\int _X\int _Y f(x,y)\,d\nu (y)\,d\mu (x) =\int _Y\int _X f(x,y)\,d\mu (x)\,d\nu (y).\]
[Definition: Region under a graph] Let \(X\) be a set and let \(f:X\to [0,\infty ]\). The region under the graph of \(f\) is \[U_f=\{(x,t):x\in X,0<t<f(x)\}.\]
(Problem 3120) Let \(E_{m,k}=f^{-1}([\frac {m}{k},\frac {m+1}{k}])\times (0,\frac {m}{k})\). Show that \(U_f=\bigcup _{k=1}^\infty \bigcup _{m=0}^{\infty } E_{m,k}\).
(Problem 3130) Let \((X,\S ,\mu )\) be a \(\sigma \)-finite measure space. Let \(f:X\to [0,\infty ]\) be a \(\S \)-measurable function. Let \(\mathcal {B}\) be the Borel sets and let \(\lambda \) denote Lebesgue measure on \(\R \). Show that \(U_f\in \S \otimes \B \).
(Problem 3140) Show that \((\mu \times \lambda )(U_f)=\int _X f\,d\mu \).
(Problem 3150) Show that \(\int _X f\,d\mu = \int _{(0,\infty )} \mu (\{x\in X:t<f(x)\})\,d\lambda (t)\).
(Problem 3160) Show that if \(\int _X f\,d\mu <\infty \) then \(\int _{(0,\infty )} \mu (\{x\in X:t<f(x)\})\,d\lambda (t)=\int _0^\infty \mu (\{x\in X:t<f(x)\})\,dt\), where the right hand integral is an improper Riemann integral.
(Problem 3170) Use the above result to give another proof of Markov’s inequality (that is, \(\mu (\{x\in X:t<f(x)\})\leq \frac {1}{t}\int _X f\,d\mu \)).
(Problem 3171) Can you do the converse, that is, use Markov’s inequality to get the above result?