Math 55003
Theory of Functions of a Real Variable I
Fall 2024

(HW 1.1) Let \(\mathcal{F}\) be a family of sets.

1. Show that \(\subseteq\) is a partial ordering on \(\mathcal{F}\).
2. When is \(A\in\mathcal{F}\) a maximal element with respect to the relation \(\subseteq\)?
3. If \(A\in\mathcal{F}\) and \(\mathcal{E}\subseteq\mathcal{F}\), when is \(A\) an upper bound for \(\mathcal{E}\) under the relation \(\subseteq\)?
4. The relation \(\supseteq\) is also a partial ordering on \(\mathcal{F}\). When is \(A\in\mathcal{F}\) a maximal element with respect to the relation \(\supseteq\)?
5. If \(A\in\mathcal{F}\) and \(\mathcal{E}\subseteq\mathcal{F}\), when is \(A\) an upper bound for \(\mathcal{E}\) under the relation \(\supseteq\)?

(HW 1.2) Let \(\mathbb{F}\) be an ordered field and let \(\mathbb{N}_{\mathbb{F}}\) be the natural numbers in \(\mathbb{F}\) (in the sense of being the intersection of all inductive sets). Show that if \(n\in\mathbb{N}_{\mathbb{F}}\) and \(n\lt f \lt n+1\), then \(f\notin \mathbb{N}_{\mathbb{F}}\). Use the definition of natural numbers in terms of inductive sets; do not invoke known facts regarding the natural numbers constructed from the Peano axioms. Hint: One approach is to start by showing that if \(n\in\mathbb{N}_{\mathbb{F}}\) and \(n\neq 1\), then \(n-1\in\mathbb{N}_{\mathbb{F}}\). (This is Problem 1.9 in your book but if you use it, I would like you to prove it.)

(HW 1.3) [revised]

1. \(F=\mathbb{Z}/7\mathbb{Z}\) is a field. Let \(\varphi:F\to\mathbb{R}\) be a ring homomorphism, that is, satisfy \(\varphi(a+b)=\varphi(a)+\varphi(b)\) for all \(a\), \(b\in F\). Show that \(\varphi(a)=0\) for all \(a\in F\).
2. Let \(F\) and \(R\) be two ordered fields. If \(n\in\mathbb{N}_F\), let \(A_n=\{k\in\mathbb{N}_F:k\leq n\}\). Let \(S=\{n\in\mathbb{N}_F:\)there is a unique injective function \(\varphi_n:A_n\to\mathbb{N}_R\) such that \(\varphi_n(1_F)=1_R\) and \(\varphi_n(k+1_F)=\varphi_n(k)+1_R\) whenever \(k\), \(k+1\in A_n\}\). Show that \(S=\mathbb{N}_F\).
3. Use the functions \(\varphi_n\) of part (b) to show that there exists a unique injective function \(\varphi:\mathbb{N}_F\to\mathbb{N}_R\).
4. Show that \(\varphi:\mathbb{N}_F\to\mathbb{N}_R\) is surjective.

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(HW 2.1) Prove that the collection of Borel sets \(\mathcal{B}\) is the smallest \(\sigma\)-algebra that contains \(\{(-\infty,a):a\in\mathbb{R}\}\). (To prove this, you need to prove not only that \(\{(-\infty,a):a\in\mathbb{R}\}\subseteq\mathcal{B}\), but also that no smaller \(\sigma\)-algebra contains \(\{(-\infty,a):a\in\mathbb{R}\}\).)

(HW 2.2) Let \((X,\mathcal{M})\) be a measurable space and let \(\mu\) be a measure on \((X,\mathcal{M})\). Suppose that \(\{E_k\}_{k=1}^\infty\) is a sequence of elements of \(\mathcal{M}\). We do not make any assumptions about disjointness or nondisjointness of the sets \(E_k\). Prove that \(\mu\left(\bigcup_{k=1}^\infty E_k\right)\leq \sum_{k=1}^\infty \mu(E_k)\).

(HW 2.3) Let \(A\), \(B\subset\mathbb{R}\) be bounded. Suppose that \(r=\inf\{|a-b|:a\in A,b\in B\}\gt 0\). Show that \(m^*(A\cup B)=m^*(A)+ m^*(B)\).

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(HW 3.1)

1. Let \(A\), \(B\subseteq\mathbb{R}\). Assume that \(\sup A\leq \inf B\). Show that \(m^*(A\cup B)=m^*(A)+m^*(B)\).
2. Let \(E\subseteq\mathbb{R}\) satisfy \(m^*(E)\lt\infty\). Define \(f:\mathbb{R}\to [0,\infty)\) by \(f(a)=m^*(E\cap (-\infty,a))\). Show that \(f\) is continuous.

(HW 3.2) Let \(E\subseteq\mathbb{R}\). Suppose that \(m^*(E)\gt0\).

1. Show that there is a bounded set \(D\) with \(D\subseteq E\) and with \(m^*(D)>0\).
2. Suppose that in addition \(m^*(E)<\infty\) and that \(\varepsilon>0\). Show that there is a bounded set \(D\) with \(D\subseteq E\) and with \(m^*(D)>m^*(E)-\varepsilon\).

(HW 3.3) Let \(E\subseteq\mathbb{R}\). Suppose that \(m^*(E)\lt\infty\).

1. Show that there is a \(G_\delta\) set \(G\) with \(E\subseteq G\) and with \(m^*(G)=m^*(E)\), even if \(E\) is not measurable.
2. Suppose that there is a \(F_\sigma\) set \(F\) with \(F\subseteq E\) that satisfies \(m^*(F)=m^*(E)\). Show that \(E\) is measurable.

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(HW 4.1) Let \(E\subseteq\mathbb{R}\). Suppose that, for all \(a\), \(b\in\mathbb{R}\) with \(a\lt b\), we have that \[b-a=m^*((a,b)\cap E)+m^*((a,b)\sim E).\] Show that \(E\) is measurable.

(HW 4.2) Let \(f:\mathbb{R}\to\mathbb{R}\) be continuous. Let \(B\) be a Borel set. Show that \(f^{-1}(B)\) is also a Borel set.

(HW 4.3) Let \(f:\mathbb{R}\to\mathbb{R}\) be continuous and strictly increasing. Let \(B\) be a Borel set. Show that \(f(B)\) is also a Borel set.

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(HW 5.1) Let \(f:[0,1]\to[0,1]\) be Lipschitz; that is, there exists a constant \(c<\infty\) such that if \(x\), \(y\in [0,1]\) then \(|f(x)-f(y)\leq c|x-y|\). Suppose that \(A\subseteq [0,1]\) is measurable. Show that \(f(A)\) is measurable. Hint: Start with the cases where \(A\) has measure zero and where \(A\) is a \(F_\sigma\)-set and then use Theorem 2.11.

(HW 5.2) Let \((X,\mathcal{A})\) be a measurable space (that is, let \(X\) be a set and let \(\mathcal{A}\) be a \(\sigma\)-algebra over \(X\)). Let \(\mu\) be a function \(\mu:\mathcal{A}\to[0,\infty]\) such that

1. \(\mu(\emptyset)=0\),
2. \(\mu(A\cup B)=\mu(A)+\mu(B)\) for all \(A\), \(B\in\mathcal{A}\) with \(A\cap B=\emptyset\),
3. If \(\{A_k\}_{k=1}^\infty\) is a sequence of sets in \(\mathcal{A}\) such that \(A_k\subseteq A_{k+1}\) for all \(k\geq 1\), then \(\mu\Bigl(\bigcup_{k=1}^\infty A_k \Bigr) =\lim_{n\to\infty}\mu(A_n)\).

(HW 5.3) Let \(E\subseteq\mathbb{R}\) be measurable and let \(f:E\to\mathbb{R}\) be a measurable function. Let \(B\subseteq\mathbb{R}\) be a Borel set. Show that \(f^{-1}(B)\) is measurable.

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(HW 6.1) Let \(E\subseteq \mathbb{R}\) be measurable. Find a necessary and sufficient condition on \(E\) such that, whenever \(f\), \(g:E\to\mathbb{R}\) are continuous functions that satisfy \(f=g\) almost everywhere on \(E\), then in fact \(f=g\) everywhere on E.

(HW 6.2) Let \(\{f_n\}_{n=1}^\infty\) be a sequence of measurable functions from \(E\) to~\(\mathbb{R}\), where \(E\subseteq\mathbb{R}\) is measurable. Let \(E_0=\{x\in E:\{f_n(x)\}_{n=1}^\infty\) converges\(\}\). Is \(E_0\) measurable? If so, prove it; if not, give a counterexample. Hint: Write \(\{x\in E:\{f_n(x)\}_{n=1}^\infty\) is Cauchy\(\}\) using unions and intersections.

(HW 6.3) 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 \(h:I\to\mathbb{R}\) and a measurable set \(F\subseteq I\) such that \(h=\chi_E\) on \(F\) and such that \(m(I\sim F)\lt\varepsilon\).

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(HW 7.1) Prove the converse of Tietze's extension theorem in \(\mathbb{R}\). That is, suppose that \(E\subseteq\mathbb{R}\) and that, if \(f:E\to\mathbb{R}\) is continous, then there exists a \(g:\mathbb{R}\to\mathbb{R}\) that is continuous and satisfies \(f=g\) on \(E\). Show that \(E\) is closed.

(HW 7.2) Let \(K\subseteq\mathbb{R}\) be compact (closed and bounded), and let \(f_n\), \(f:K\to\mathbb{R}\). Suppose that \(f\) and each \(f_n\) is continuous, that \(f_n\leq f_{n+1}\) for all \(n\), and that \(f_n\to f\) pointwise on \(K\). Show that \(f_n\to f\) uniformly. (Note that the assumption that \(f\) is continuous is necessary; the result is false if \(f\) fails to be continuous!)

(HW 7.3) Let \(E\subset\mathbb{R}\) be measurable with \(m(E)\lt\infty\), and let \(f:E\to[-M,M]\) be a bounded measurable function. Let \(A\subset E\) be measurable. Show that \(\int_A f=\int_E f\chi_A.\) (Note that we used this problem in the proof of Corollary 4.6, and so you may not use Corollary 4.6 in your solution.)

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(HW 8.1) Let \(E\subseteq\mathbb{R}\) be a measurable set and let \(f:E\to[0,\infty]\) be a nonnegative measurable function. Show that \[\int_E f=\sup\biggl\{\int_E \varphi:\varphi\text{ is simple and } 0\leq \varphi\leq f\text{ on }E.\biggr\}\] In particular, need we require that \(\varphi\) be of finite support?

(HW 8.2) Let \(a\), \(b\in\mathbb{R}\) with \(a\lt b\) and let \(f:[a,b]\to[0,\infty]\) be nonnegative and integrable. Let \(\varepsilon\gt 0\). Show that there is a step function \(h:[a,b]\to [0,\infty)\) such that \(\int_{[a,b]} |f-h|\lt\varepsilon\).

(HW 8.3) Let \(f:[1,\infty)\to\mathbb{R}\) be measurable. Suppose that, if \(K\subset [1,\infty)\) is compact, then \(f\) is bounded on K. Define \(a_n=\int_n^{n+1} f\).

1. Suppose that \(f\) is integrable on \([1,\infty)\). Is it necessarily true that \(\sum_{n=1}^\infty a_n\) converges absolutely?
2. Suppose that \(f\) is integrable on \([1,\infty)\). Is it necessarily true that \(\sum_{n=1}^\infty a_n\) converges?
3. Suppose that \(\sum_{n=1}^\infty a_n\) converges absolutely. Is it necessarily true that \(f\) is integrable on \([1,\infty)\)?
4. Suppose that \(\sum_{n=1}^\infty a_n\) converges. Is it necessarily true that \(f\) is integrable on \([1,\infty)\)?

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(HW 9.1) Let \(E\subseteq\mathbb{R}\) be a measurable set and let \(f_n\), \(g_n\), \(f\), and \(g\) be measurable functions on \(E\) and suppose that \(g_n\), \(g\) are nonnegative. Suppose that

• \(|f_n(x)|\leq |g_n(x)|\) for all \(x\in E\) and all \(n\in\mathbb{N}\),
• \(f_n\to f\) pointwise almost everywhere on \(E\),
• \(g_n\to g\) pointwise almost everywhere on \(E\),
• \(\lim_{n\to\infty} \int_E g_n=\int_E g\),
• \(\int_E g\lt\infty\).

(HW 9.2) Let \(f\) be integrable over \(\mathbb{R}\) and let \(\varepsilon\gt0\). Verify the following three facts.

1. There is a simple function \(\eta\) with finite support and such that \(\int_{\mathbb{R}}|f-\eta|<\varepsilon\).
2. There is a step function \(s\) which is zero outside of some closed bounded interval and such that \(\int_{\mathbb{R}}|f-s|<\varepsilon\).
3. There is a continuous function \(g\) which is zero outside of some closed bounded interval and such that \(\int_{\mathbb{R}}|f-g|<\varepsilon\).

(HW 9.3) Let \(E\subseteq\mathbb{R}\) be measurable. Let \(\{h_n\}_{n=1}^\infty\) be a sequence of nonnegative functions defined on \(E\) each of which is integrable on \(E\). Suppose that \(h_n\to 0\) pointwise almost everywhere on \(E\). Show that \[\lim_{n\to\infty} \int_E h_n=0\] if and only if \(\{h_n:n\in\mathbb{N}\}\) is both uniformly integrable and tight over \(E\). Note that we will use this problem in the proof of Corollary 5.5, so you may not use Corollary 5.5 in the solution to this problem.

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(HW 10.1) Let \(E\subseteq\mathbb{R}\) be measurable, and let \(f_n\), \(f\), and \(g\) be measurable functions on \(E\) that are finite almost everywhere on \(E\). Suppose that \(f_n\to f\) in measure and that \(f_n\to g\) in measure. Show that \(f(x)=g(x)\) for almost every \(x\in E\).

(HW 10.2) Show that Lebesgue’s dominated convergence theorem is still valid if we replace “convergence pointwise almost everywhere” by “convergence in measure”.

(HW 10.3) Let \(E\subseteq\mathbb{R}\) be measurable and let \(f:E\to[0,\infty]\) be a nonnegative function. Show that \(\int_E f=\int_0^\infty m(\{x\in E:f(x)\gt\lambda\})\,d\lambda\) (where the integral is in the sense of Section 4.3).

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(HW 11.1) Let \(I\subseteq\mathbb{R}\) be a nonempty interval and let \(f:I\to \mathbb{R}\) be a non-constant nondecreasing function. Let \(J=(\inf \{f(x):x\in I\},\sup\{f(x):x\in I\}\). Define \(g:J\to I\) by \(g(y)=\sup\{x:f(x)\leq y\}\). Show that \(g(f(x))=x\) whenever \(g\) is continuous at \(f(x)\) and that \(f(g(y))=y\) whenever \(f\) is continuous at \(g(y)\).

(HW 11.2) Let \(E\subset\mathbb{R}\) be a measurable set with \(m(E)\lt\infty\). Let \(\mathcal{C}\) be a (possibly infinite) collection of closed bounded intervals in \(\mathbb{R}\) such that \(E\subseteq\bigcup_{I\in \mathcal{C}} I\). Suppose that there is an \(r\geq 0\) such that \(2r\leq \ell(I) \leq 3r\) for all \(I\in \mathcal{C}\). Using the fact that \(\mathbb{R}\) is separable, and without using Zorn's lemma (or Lecture Problem 1711), show that there is a countable subcollection \(\mathcal{S}\subseteq \mathcal{C}\) such that \[E\subseteq \bigcup_{I\in \mathcal{S}} 5I\] and such that, if \(I\), \(J\in \mathcal{S}\), then \(I\cap J=\emptyset\).

(HW 11.3) Let \(\mathcal{S}\) be a nonempty collection of pairwise-disjoint closed intervals in \(\mathbb{R}\). Suppose that \(r=\inf\{\ell(I):I\in\mathcal{S}\}\gt 0\). Show that \(\bigcup_{I\in\mathcal{S}} I\) is closed.

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(HW 12.1) Let \(E\subseteq\mathbb{R}\). Show that \(m^*(E)=0\) if and only if there exists a countable collection of open intervals \(\{I_k\}_{k=1}^\infty\) such that

• \(\sum_{k=1}^\infty \ell(I_k)\lt\infty\).
• If \(x\in E\), then the set \(\{k\in\mathbb{N}:x\in I_k\}\) is infinite.

(HW 12.2) Let \(a\), \(b\in\mathbb{R}\) with \(a\lt b\) and let \(E\subset (a,b)\) satisfy \(m(E)=0\). Show that there is a nondecreasing function \(f:(a,b)\to\mathbb{R}\) such that \(f\) is discontinuous not differentiable at every point \(e\in E\). Hint: Let \(\{I_k\}_{k=1}^\infty\) be as in Problem 12.1 and let \[f(x)=\sum_{k=1}^\infty \ell(I_k\cap (-\infty,x)).\]

(HW 12.3) 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.

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(HW 13.1) Let \([a,b]\subset\mathbb{R}\) be a closed and bounded interval and let \(f\), \(g:[a,b]\to\mathbb{R}\). Suppose that \(f\) and \(g\) are both absolutely continuous over \([a,b]\).

1. Show that \(f+g\) is absolutely continuous over \([a,b]\).
2. Show that \(fg\) is absolutely continuous over \([a,b]\).

(HW 13.2) Suppose that \([a,b]\subset\mathbb{R}\) is a closed bounded interval and that \(f:[a,b]\to\mathbb{R}\) is continuous and increasing nondecreasing. Show that \(f\) is absolutely continuous if and only if, for every \(\varepsilon\gt0\), there is a \(\delta\gt 0\) such that if \(E\subseteq[a,b]\) is measurable and \(m(E)\lt\delta\), then \(m^*(f(E))\lt \varepsilon\). You may assume that Problem 6.38 is correct.

(HW 13.3) Let \([a,b]\subset\mathbb{R}\) be a closed and bounded interval and let \(f:[a,b]\to\mathbb{R}\). Suppose that \(f\) is absolutely continuous. Show that \(f\) is Lipschitz if and only if there is a \(c\gt0\) such that \(|f'(x)|\lt c\) for almost every \(x\in [a,b]\). What can you say about \(c\) and the Lipschitz constant of \(f\)?

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(HW 14.1) Let \(I\subseteq\mathbb{R}\) be an open interval and let \(\varphi:I\to\mathbb{R}\).

1. Show that \(\varphi\) is convex if and only if \[\varphi\Bigl(\sum_{k=1}^n \lambda_k x_k\Bigr) \leq \sum_{k=1}^n \lambda_k \varphi(x_k)\] whenever \(n\in\mathbb{N}\), \(x_k\in I\) for all \(1\leq k\leq n\), \(\lambda_k\geq 0\) for all \(1\leq k\leq n\), and \(\sum_{k=1}^n \lambda_k=1\).
2. Use the previous part to give a direct proof that if \(\varphi:\mathbb{R}\to\mathbb{R}\) is convex and \(f:[0,1]\to\mathbb{R}\) is simple, then \[\varphi\biggl(\int_0^1 f\biggr)\leq \int_0^1 (\varphi\circ f).\] Do not use Jensen's inequality (or its proof) as stated in the text.

(HW 14.2) Let \(C=C[0,1]\) denote the real vector space of continuous functions over \([0,1]\).

1. Show that \(\|f\|_1=\|f\|_{L^1([0,1])}\) is a norm on \(C\).
2. Show that \(\|f\|_\infty=\max_{[0,1]}|f|\) is a norm on \(C\).
3. Show that \(\|f\|_1\leq\|f\|_\infty\) for all \(f\in C\).
4. Show that there does not exist a constant \(c\) such that \(\|f\|_\infty\leq c\|f\|_1\) for all \(f\in C\).

(HW 14.3) Let \(0\lt p\lt 1\) and let \(E\subseteq\mathbb{R}\) be measurable with \(m(E)\gt 0\).

1. Recall that \[\|f\|_{L^p(E)} = \biggl(\int_E |f|^p\biggr)^{1/p}.\] We define \(\|\tilde f\|_{L^p(E)}\), for \(\tilde f\in L^p(E)\) (that is, for \(\tilde f\) an equivalence class of functions under the relation \(\cong\)), in the obvious way. Show that this is not a norm on \(L^p(E)\).
2. Let \(f\), \(g\in L^p(E)\). Show that \(\|f+g\|_{L^p(E)}^p\leq \|f\|_{L^p(E)}^p+\|g\|_{L^p(E)}^p.\) Hint: Suppose that \(x\) is a nonnegative real number. Show that \((1+x)^p\leq 1+x^p\) by analyzing the derivative of \(\varphi(x)=1+x^p-(1+x)^p\).

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