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

# 8.1.E: Problems on Measurable and Elementary Functions in $$(S, \mathcal{M})$$

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

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

Fill in all proof details in Corollaries 2 and 3 and Theorems 1 and 2.

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

Show that $$\mathcal{P}^{\prime} \cap P^{\prime \prime}$$ is as stated at the end of Definition 2.

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

Given $$A \subseteq S$$ and $$f, f_{m}: S \rightarrow\left(T, \rho^{\prime}\right), m=1,2, \ldots,$$ let
$H=A\left(f_{m} \rightarrow f\right)$
and
$A_{m n}=A\left(\rho^{\prime}\left(f_{m}, f\right)<\frac{1}{n}\right) .$
Prove that
(i) $$H=\bigcap_{n=1}^{\infty} \bigcup_{k=1}^{\infty} \bigcap_{m=k}^{\infty} A_{m n} ;$$
(ii) $$H \in \mathcal{M}$$ if all $$A_{m n}$$ are in $$\mathcal{M}$$ and $$\mathcal{M}$$ is a $$\sigma$$-ring.
[Hint: $$x \in H$$ iff
$(\forall n)(\exists k)(\forall m \geq k) \quad x \in A_{m n} .$
Why?]

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

Do Problem 3 for $$T=E^{*}$$ and $$f=\pm \infty$$ on $$H$$.
[Hint: If $$\left.f=+\infty, A_{m n}=A\left(f_{m}>n\right) \cdot\right]$$

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

$$\Rightarrow 4$$. Let $$f: S \rightarrow T$$ be $$\mathcal{M}$$-elementary on $$A,$$ with $$\mathcal{M}$$ a $$\sigma$$ -ring in $$S .$$ Show the following.
(i) $$A(f=a) \in \mathcal{M}, A(f \neq a) \in \mathcal{M}$$.
(ii) If $$T=E^{*},$$ then
$$A(f<a), A(f \geq a), A(f>a),$$ and $$A(f \geq a)$$
are in $$\mathcal{M},$$ too.
(iii) $$(\forall B \subseteq T) A \cap f^{-1}[B] \in \mathcal{M}$$.
[Hint: If
$A=\bigcup_{i-1}^{\infty} A_{i}$
and $$\left.f=a_{i} \text { on } A_{i}, \text { then } A(f=a) \text { is the countable union of those } A_{i} \text { for which } a_{i}=a .\right]$$

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

Do Problem $$4(\mathrm{i})$$ for measurable $$f$$.
[Hint: If $$f=\lim f_{m}$$ for elementary maps $$f_{m},$$ then
$H=A(f=a)=A\left(f_{m} \rightarrow a\right) .$
Express $$H$$ as in Problem $$3,$$ with
$A_{m n}=A\left(h_{m}<\frac{1}{n}\right) ,$
where $$h_{m}=\rho^{\prime}\left(f_{m}, a\right)$$ is elementary. (Why?) Then use Problems $$4(\text { ii) and } 3(\text { ii }) .]$$

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

$$\Rightarrow 6$$. Given $$f, g: S \rightarrow\left(T, \rho^{\prime}\right),$$ let $$h=\rho^{\prime}(f, g),$$ i.e.,
$h(x)=\rho^{\prime}(f(x), g(x)) .$
Prove that if $$f$$ and $$g$$ are elementary, simple, or measurable on $$A,$$ so is $$h .$$
[Hint: Argue as in Theorem 1. Use Theorem $$4 \text { in Chapter } 3, §15 .]$$

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

$$\Rightarrow 7$$. $$\left. \text { A set }\left.B \subseteq\left(T, \rho^{\prime}\right) \text { is called separable (in } T\right) \text { iff } B \subseteq \overline{D} \text { (closure of } D\right)$$ for a countable set $$D \subseteq T$$.
Prove that if $$f: S \rightarrow T$$ is $$\mathcal{M}$$-measurable on $$A,$$ then $$f[A]$$ is separable in $$T .$$
[Hint: $$f=\lim f_{m}$$ for elementary maps $$f_{m} ;$$ say,
$f_{m}=a_{m i} \text { on } A_{m i} \in \mathcal{M}, \quad i=1,2, \ldots$
Let $$D$$ consist of all $$a_{m \mathrm{i}}(m, i=1,2, \ldots) ;$$ so $$D$$ is countable (why?) and $$D \subseteq T$$.
Verify that
$(\forall y \in f[A])(\exists x \in A) \quad y=f(x)=\lim f_{m}(x) ,$
with $$f_{m}(x) \in D .$$ Hence
$(\forall y \in f[A]) \quad y \in \overline{D} ,$
by Theorem $$3 \text { of Chapter } 3, §16 .]$$

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

$$\Rightarrow 8$$. Continuing Problem $$7,$$ prove that if $$B \subseteq \overline{D}$$ and $$D=\left\{q_{1}, q_{2}, \ldots\right\},$$ then
$(\forall n) \quad B \subseteq \bigcup_{i=1}^{\infty} G_{q_{i}}\left(\frac{1}{n}\right) ,$
[Hint: If $$p \in B \subseteq \overline{D},$$ any $$G_{p}\left(\frac{1}{n}\right)$$ contains some $$q_{1} \in D ;$$ so
$\rho^{\prime}\left(p, q_{i}\right)<\frac{1}{n}, \text { or } p \in G_{q_{i}}\left(\frac{1}{n}\right) .$
Thus
$\left.(\forall p \in B) \quad p \in \bigcup_{i-1}^{\infty} G_{q_{i}}\left(\frac{1}{n}\right) \cdot\right]$

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

Prove Corollaries 2 and 3 and Theorems 1 and $$2,$$ assuming that $$\mathcal{M}$$ is a semiring only.

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

Do Problem 4 for $$\mathcal{M}$$-simple maps, assuming that $$\mathcal{M}$$ is a ring only.