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

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.