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


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

Fill in all proof details in Corollaries 1 to 4.

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

Verify Notes 3 and 4.

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

Prove Theorems 1 and 2 in §1 and Theorem 2 in §2, for almost measurable functions.

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

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 $$\left.f: S \rightarrow E^{n}\left(C^{n}\right), \text { use Theorem } 2 \text { of } §1 .\right]$$

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

*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 $$13 \text { in } §2 .]$$

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

*5. Prove Theorem 1 for $$f: S \rightarrow\left(T, \rho^{\prime}\right),$$ assuming that $$f[A]$$ is separable in $$T .$$

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

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 $$A .$$

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

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 $$m_{A}$$.

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

(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 $$\xi 1$$.]

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

*9. Fill in all proof details in Lemma 4.

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

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.

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

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
$(\forall \sigma>0) \quad \lim _{n \rightarrow \infty} m A\left(\rho^{\prime}\left(f, f_{n}\right) \geq \sigma\right)=0 .$

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

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). $$]$$

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

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 $$n \rightarrow \infty .]$$

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

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.]