8.1.E: Problems on Measurable and Elementary Functions in \((S, \mathcal{M})\)
Fill in all proof details in Corollaries 2 and 3 and Theorems 1 and 2.
Show that \(\mathcal{P}^{\prime} \cap P^{\prime \prime}\) is as stated at the end of Definition 2.
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?]
Do Problem 3 for \(T=E^{*}\) and \(f=\pm \infty\) on \(H\).
[Hint: If \(\left.f=+\infty, A_{m n}=A\left(f_{m}>n\right) \cdot\right]\)
\(\Rightarrow 4\). 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 \(\left.f=a_{i} \text { on } A_{i}, \text { then } A(f=a) \text { is the countable union of those } A_{i} \text { for which } a_{i}=a .\right]\)
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 \(4(\text { ii) and } 3(\text { ii }) .]\)
\(\Rightarrow 6\). 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 \(4 \text { in Chapter } 3, §15 .]\)
\(\Rightarrow 7\). \(\left. \text { A set }\left.B \subseteq\left(T, \rho^{\prime}\right) \text { is called separable (in } T\right) \text { iff } B \subseteq \overline{D} \text { (closure of } D\right)\) 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 \(3 \text { of Chapter } 3, §16 .]\)
\(\Rightarrow 8\). 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
\[
\left.(\forall p \in B) \quad p \in \bigcup_{i-1}^{\infty} G_{q_{i}}\left(\frac{1}{n}\right) \cdot\right]
\]
Prove Corollaries 2 and 3 and Theorems 1 and \(2,\) assuming that \(\mathcal{M}\) is a semiring only.
Do Problem 4 for \(\mathcal{M}\)-simple maps, assuming that \(\mathcal{M}\) is a ring only.