$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 8.2.E: Problems on Measurable Functions in $$(S, \mathcal{M}, m)$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Exercise $$\PageIndex{1}$$

In Theorem $$1,$$ give the details in proving the equivalence of $$\left(i^{*}\right)-\left(i v^{*}\right)$$.

Prove Note 1.

## Exercise $$\PageIndex{2'}$$

Prove that $$f=f^{+}-f^{-}$$ and $$|f|=f^{+}+f^{-}$$.

## Exercise $$\PageIndex{3}$$

Complete the proof of Theorem $$2,$$ in detail.

## Exercise $$\PageIndex{4}$$

$$\Rightarrow 4$$. Prove Theorem 3.
[Hint: By our conventions, $$A(f \geq g)=A(f-g \geq 0)$$ even if $$g$$ or $$f$$ is $$\pm \infty$$ for some $$x \in A .$$ (Verify all cases!) By Theorems 1 and $$2, A(f-g \geq 0) \in \mathcal{M} ;$$ so $$A(f \geq g) \in \mathcal{M}, \text { and } A(f<g)=A-A(f \geq g) \in \mathcal{M} . \text { Proceed. }]$$

## Exercise $$\PageIndex{5}$$

Show that the measurability of $$|f|$$ does not imply that of $$f$$.
[Hint: Let $$f=1$$ on $$Q$$ and $$f=-1$$ on $$A-Q$$ for some $$Q \notin \mathcal{M}(Q \subset A) ;$$ e.g., use $$Q$$ of Problem $$6 \text { in Chapter } 7, § 8 .]$$

## Exercise $$\PageIndex{6}$$

$$\Rightarrow 6$$. Show that a function $$f \geq 0$$ is measurable on $$A$$ iff $$f_{m} \nearrow f$$ (pointwise) on $$A$$ for some finite simple maps $$f_{m} \geq 0,\left\{f_{m}\right\} \uparrow$$.
[Hint: Modify the proof of Lemma $$2,$$ setting $$H_{m}=A(f \geq m)$$ and $$f_{m}=m$$ on $$H_{m}$$, and defining the $$A_{m k}$$ for $$1 \leq k \leq m 2^{m}$$ only.]

## Exercise $$\PageIndex{7}$$

$$\Rightarrow 7$$. Prove Theorem 3 in $$§ 1 .$$
[Outline: By Problems 7 and 8 in $$\xi 1,$$ there are $$q_{i} \in T$$ such that
$(\forall n) \quad f[A] \subseteq \bigcup_{i=1}^{\infty} G q_{i}\left(\frac{1}{n}\right) .$
Set
$A_{n i}=A \cap f^{-1}\left[G_{q_{i}}\left(\frac{1}{n}\right)\right] \in \mathcal{M}$
by Corollary $$2 ;$$ so $$\rho^{\prime}\left(f(x), q_{i}\right)<\frac{1}{n}$$ on $$A_{n i}$$.
By Corollary 1 in Chapter 7, §1
$A=\bigcup_{i=1}^{\infty} A_{n i}=\bigcup_{i=1}^{\infty} B_{n i}(\text {disjoint})$
for some sets $$B_{n i} \in \mathcal{M}, B_{n i} \subseteq A_{n i} .$$ Now define $$g_{n}=q_{i}$$ on $$B_{n i} ;$$ so $$\rho^{\prime}\left(f, g_{n}\right)<\frac{1}{n}$$ on each $$\left.B_{n i}, \text { hence on } A . \text { By Theorem } 1 \text { in Chapter } 4, §12, g_{n} \rightarrow f \text { (uniformly) on } A .\right]$$

## Exercise $$\PageIndex{8}$$

$$\Rightarrow 8$$. Prove that $$f: S \rightarrow E^{1}$$ is $$\mathcal{M}$$ -measurable on $$A$$ iff $$A \cap f^{-1}[B] \in \mathcal{M}$$ for every Borel set $$B \text { (equivalently, for every open set } B)$$ in $$E^{1} .$$ (In the case $$f: S \rightarrow E^{*},$$ add: "and for $$B=\{\pm \infty\}.$$ ")
[Outline: Let
$\mathcal{R}=\left\{X \subseteq E^{1} | A \cap f^{-1}[X] \in \mathcal{M}\right\} .$
Show that $$\mathcal{R}$$ is a $$\sigma$$ -ring in $$E^{1}$$.
Now, by Theorem $$1,$$ if $$f$$ is measurable on $$A, \mathcal{R}$$ contains all open intervals; for
$A \cap f^{-1}[(a, b)]=A(f>a) \cap A(f<b) .$
Then by Lemma 2 of Chapter $$7, 2, \mathcal{R} \supseteq \mathcal{G},$$ hence $$\mathcal{R} \supseteq \mathcal{B} .$$ (Why?)
Conversely, if so,
$\left.(a, \infty) \in \mathcal{R} \Rightarrow A \cap f^{-1}[(a, \infty)] \in \mathcal{M} \Rightarrow A(f>a) \in \mathcal{M} .\right]$

## Exercise $$\PageIndex{9}$$

$$\Rightarrow 9$$. Do Problem 8 for $$f: S \rightarrow E^{n}$$.
[Hint: If $$f=\left(f_{1}, \ldots, f_{n}\right)$$ and $$B=(\overline{a}, \overline{b}) \subset E^{n},$$ with $$\bar{a}=\left(a_{1}, \ldots, a_{n}\right)$$ and $$\bar{b}=$$ $$\left(b_{1}, \ldots, b_{n}\right),$$ show that
$f^{-1}[B]=\bigcap_{k=1}^{n} f_{k}^{-1}\left[\left(a_{k}, b_{k}\right)\right] .$
Apply Problem 8 to each $$f_{k}: S \rightarrow E^{1}$$ and use Theorem 2 in §1. Proceed as in Problem $$8 .]$$

## Exercise $$\PageIndex{10}$$

Do Problem 8 for $$f: S \rightarrow C^{n},$$ treating $$C^{n}$$ as $$E^{2 n}$$.

## Exercise $$\PageIndex{11}$$

Prove that $$f: S \rightarrow\left(T, \rho^{\prime}\right)$$ is measurable on $$A$$ in $$(S, \mathcal{M})$$ iff
(i) $$A \cap f^{-1}[G] \in \mathcal{M}$$ for every open globe $$G \subseteq T,$$ and
(ii) $$f[A]$$ is separable in $$T(\text { Problem } 7 \text { in } § 1) .$$
[Hint: If so, proceed as in Problem $$7 \text { (without assuming measurability of } f)$$ to show that $$f=\lim g_{n}$$ for some elementary maps $$g_{n}$$ on $$A .$$ For the converse, use Problem 7 in $$§ 1$$ and Corollary 2 in $$§ 2$$.]

## Exercise $$\PageIndex{12}$$

(i) Show that if all of $$T \text { is separable (Problem } 7 \text { in } § 1),$$ there is a sequence of globes $$G_{k} \subseteq T$$ such that each nonempty open set
$$B \subseteq T$$ is the union of some of these $$G_{k}$$.
(ii) Show that $$E^{n}$$ and $$C^{n}$$ are separable.
[Hints: (i) Use the $$G_{q_{i}}\left(\frac{1}{n}\right)$$ of Problem 8 in $$§ 1,$$ putting them in one sequence.
(ii) Take $$\left.D=R^{n} \subset E^{n} \text { in Problem } 7 \text { of } § 1 .\right]$$

## Exercise $$\PageIndex{13}$$

Do Problem 11 with "globe $$G \subseteq T^{\prime \prime}$$ replaced by "Borel set $$B \subseteq T$$."
[Hints: Treat $$f$$ as $$f: A \rightarrow T^{\prime}, T^{\prime}=f[A],$$ noting that
$A \cap f^{-1}[B]=A \cap f^{-1}\left[B \cap T^{\prime}\right] .$
By Problem $$12,$$ if $$B \neq \emptyset$$ is open in $$T,$$ then $$B \cap T^{\prime}$$ is a countable union of "globes" $$G_{q} \cap T^{\prime}$$ in $$\left(T^{\prime}, \rho^{\prime}\right) ;$$ see Theorem 4 in Chapter $$3, § 12 .$$ Proceed as in Problem $$8,$$ replacing $$E^{1}$$ by $$T$$.]

## Exercise $$\PageIndex{14}$$

A map $$g:\left(T, \rho^{\prime}\right) \rightarrow\left(U, \rho^{\prime \prime}\right)$$ is said to be of Baire class $$0\left(g \in \mathrm{B}_{0}\right)$$ on $$D \subseteq T$$ iff $$g$$ is relatively continuous on $$D .$$ Inductively, $$g$$ is of Baire class $$n\left(g \in \mathbf{B}_{n}, n \geq 1\right)$$ iff $$g=\lim g_{m}$$ (pointwise) on $$D$$ for some maps $$g_{m} \in \mathbf{B}_{n-1}$$. Show by induction that Corollary 4 in $$§ 1$$ holds also if $$g \in \mathbf{B}_{n}$$ on $$f[A]$$ for some $$n .$$