8.3.E: Problems on Measurable Functions in \((S, \mathcal{M}, m)\)
Fill in all proof details in Corollaries 1 to 4.
Verify Notes 3 and 4.
Prove Theorems 1 and 2 in §1 and Theorem 2 in §2, for almost measurable functions.
Prove Note 2.
[Hint: If \(f: S \rightarrow E^{*}\) is \(\mathcal{M}\)-measurable on \(B=A-Q(m Q=0, Q \subseteq A),\) then \(A=B \cup Q\) and
\[
\left(\forall a \in E^{*}\right) \quad A(f>a)=B(f>a) \cup Q(f>a) .
\]
Here \(B(f>a) \in \mathcal{M}\) by Theorem 1 in §2, and \(Q(f>a) \in \mathcal{M}\) if \(m\) is complete. For \(\left.f: S \rightarrow E^{n}\left(C^{n}\right), \text { use Theorem } 2 \text { of } §1 .\right]\)
*4. Show that if \(m\) is complete and \(f: S \rightarrow\left(T, \rho^{\prime}\right)\) is \(m\)-measurable on \(A\) with \(f[A]\) separable in \(T,\) then \(f\) is \(\mathcal{M}\) -measurable on \(A .\)
[Hint: Use Problem \(13 \text { in } §2 .]\)
*5. Prove Theorem 1 for \(f: S \rightarrow\left(T, \rho^{\prime}\right),\) assuming that \(f[A]\) is separable in \(T .\)
Given \(f_{n} \rightarrow f(\text { a.e. })\) on \(A,\) prove that \(f_{n} \rightarrow g(\text { a.e. })\) on \(A\) iff \(f=g(\text { a.e. })\) on \(A .\)
Given \(A \in \mathcal{M}\) in \((S, \mathcal{M}, m),\) let \(m_{A}\) be the restriction of \(m\) to
\[
\mathcal{M}_{A}=\{X \in \mathcal{M} | X \subseteq A\} .
\]
Prove that
(i) \(\left.\left(A, \mathcal{M}_{A}, m_{A}\right) \text { is a measure space (called a subspace of }(S, \mathcal{M}, m)\right)\);
(ii) if \(m\) is complete, topological, \(\sigma\)-finite or (strongly) regular, so is \(m_{A}\).
(i) Show that if \(D \subseteq K \subseteq\left(T, \rho^{\prime}\right),\) then the closure of \(D\) in the subspace \(\left(K, \rho^{\prime}\right)\) is \(K \cap \bar{D},\) where \(\bar{D}\) is the closure of \(D\) in \(\left(T, \rho^{\prime}\right) .\)
[Hint: Use Problem \(11 \text { in Chapter } 3, §16 .]\)
(ii) Prove that if \(B \subseteq K\) and if \(B\) is separable in \(\left(T, \rho^{\prime}\right),\) it is so in \(\left(K, \rho^{\prime}\right) .\)
[Hint: Use Problem 7 from \(\xi 1\).]
*9. Fill in all proof details in Lemma 4.
Simplify the proof of Theorem 2 for the case \(m A<\infty .\)
[Outline: (i) First, let \(f\) be elementary, with \(f=a_{i}\) on \(A_{i} \in \mathcal{M}, A=\cup_{i} A_{i}\) (disjoint), \(\sum m A_{i}=m A<\infty\).
Given \(\varepsilon>0\)
\[
(\exists n) \quad m A-\sum_{i=1}^{n} m A_{i}<\frac{1}{2} \varepsilon .
\]
Each \(A_{i}\) has a closed subset \(F_{i} \in \mathcal{M}\) with \(m\left(A_{i}-F_{i}\right)<\varepsilon / 2 n .\) (Why?) Now use Problem 17 in Chapter 4, §8, and set \(F=\bigcup_{i=1}^{n} F_{i} .\)
(ii) If \(f\) is \(\mathcal{M}\) -measurable on \(H=A-Q, m Q=0,\) then by Theorem 3 in \(\xi 1,\)
\(f_{n} \rightarrow f\) (uniformly) on \(H\) for some elementary maps \(f_{n} .\) By \((i),\) each \(f_{n}\) is relatively continuous on a closed \(\mathcal{M}\)-set \(F_{n} \subseteq H,\) with \(m H-m F_{n}<\varepsilon / 2^{n} ;\) so all \(f_{n}\) are relatively continuous on \(F=\bigcap_{n=1}^{\infty} F_{n} .\) Show that \(F\) is the required set.
Given \(f_{n}: S \rightarrow\left(T, \rho^{\prime}\right), n=1,2, \ldots,\) we say that
(i) \(f_{n} \rightarrow f\) almost uniformly on \(A \subseteq S\) iff
\[
(\forall \delta>0)(\exists D \in \mathcal{M} | m D<\delta) \quad f_{n} \rightarrow f(\text {uniformly}) \text { on } A-D ;
\]
(ii) \(f_{n} \rightarrow f\) in measure on \(A\) iff
\[
\begin{aligned}(\forall \delta, \sigma>0)(\exists k)(\forall n>k)\left(\exists D_{n} \in \mathcal{M} | m D_{n}<\delta\right) \\ \rho^{\prime}\left(f, f_{n}\right)<\sigma \text { on } A-D_{n} . \end{aligned}
\]
Prove the following.
(a) \(f_{n} \rightarrow f\) (uniformly) implies \(f_{n} \rightarrow f\) (almost uniformly), and the latter implies both \(f_{n} \rightarrow f\left(\text { in measure) and } f_{n} \rightarrow f(a . e .) .\right.\)
(b) Given \(f_{n} \rightarrow f\) (almost uniformly), we have \(f_{n} \rightarrow g\) (almost uniformly) iff \(f=g(\text { a.e. }) ;\) similarly for convergence in measure.
(c) If \(f\) and \(f_{n}\) are \(\mathcal{M}\) -measurable on \(A,\) then \(f_{n} \rightarrow f\) in measure on \(A\) iff
\[
(\forall \sigma>0) \quad \lim _{n \rightarrow \infty} m A\left(\rho^{\prime}\left(f, f_{n}\right) \geq \sigma\right)=0 .
\]
Assuming that \(f_{n}: S \rightarrow\left(T, \rho^{\prime}\right)\) is \(m\) -measurable on \(A\) for \(n=1,2, \ldots,\) that \(m A<\infty,\) and that \(f_{n} \rightarrow f(a . e .)\) on \(A,\) prove the following.
(i) Lebesgue's theorem: \(f_{n} \rightarrow f\) (in measure) on \(A\) (see Problem 11 ).
(ii) Egorov's theorem: \(f_{n} \rightarrow f\) (almost uniformly) on \(A\).
[Outline: (i) \(\left.f_{n} \text { and } f \text { are } \mathcal{M} \text { -measurable on } H=A-Q, m Q=0 \text { (Corollary } 1\right),\) with \(f_{n} \rightarrow f\) (pointwise) on \(H .\) For all \(i, k,\) set
\[
H_{i}(k)=\bigcap_{n=i}^{\infty} H\left(\rho^{\prime}\left(f_{n}, f\right)<\frac{1}{k}\right) \in \mathcal{M}
\]
by Problem 6 in \(\text { §1. Show that ( } \forall k) H_{i}(k) \nearrow H\); hence
\[
\lim _{i \rightarrow \infty} m H_{i}(k)=m H=m A<\infty ;
\]
so
\[
(\forall \delta>0)(\forall k)\left(\exists i_{k}\right) \quad m\left(A-H_{i_{k}}(k)\right)<\frac{\delta}{2^{k}} ,
\]
proving \((\mathrm{i}),\) since
\[
\left(\forall n>i_{k}\right) \quad \rho^{\prime}\left(f_{n}, f\right)<\frac{1}{k} \text { on } H_{i_{k}}(k)=A-\left(A-H_{i_{k}}(k)\right) .
\]
(ii) Continuing, set \((\forall k) D_{k}=H_{i_{k}}(k)\) and
\[
D=A-\bigcap_{k=1}^{\infty} D_{k}=\bigcup_{k=1}^{\infty}\left(A-D_{k}\right) .
\]
Deduce that \(D \in \mathcal{M}\) and
\[
m D \leq \sum_{k=1}^{\infty} m\left(A-H_{i_{k}}(k)\right)<\sum_{k=1}^{\infty} \frac{\delta}{2^{k}}=\delta .
\]
Now, from the definition of the \(H_{i}(k),\) show that \(f_{n} \rightarrow f\) (uniformly) on \(A-D,\) proving (ii). \(]\)
Disprove the converse to Problem \(12(\mathrm{i})\).
[Outline: Assume that \(A=[0,1) ;\) for all \(0 \leq k\) and all \(0 \leq i<2^{k},\) set
\[
g_{i k}(x)=\left\{\begin{array}{ll}{1} & {\text { if } \frac{i-1}{2^{k}} \leq x<\frac{i}{2^{k}}} \\ {0} & {\text { otherwise }}\end{array}\right.
\]
Put the \(g_{i k}\) in a single sequence by
\[
f_{2^{k}+i}=g_{i k} .
\]
Show that \(f_{n} \rightarrow 0\) in L measure on \(A,\) yet for no \(x \in A\) does \(f_{n}(x)\) converge as \(n \rightarrow \infty .]\)
Prove that if \(f: S \rightarrow\left(T, \rho^{\prime}\right)\) is \(m\) -measurable on \(A\) and \(g: T \rightarrow\left(U, \rho^{\prime \prime}\right)\) is relatively continuous on \(f[A],\) then \(g \circ f: S \rightarrow\left(U, \rho^{\prime \prime}\right)\) is \(m\)-measurable on \(A .\)
[Hint: Use Corollary 4 in §1.]