8.2.E: Problems on Measurable Functions in \((S, \mathcal{M}, m)\)
- Page ID
- 25010
In Theorem \(1,\) give the details in proving the equivalence of \(\left(i^{*}\right)-\left(i v^{*}\right)\).
Prove Note 1.
Prove that \(f=f^{+}-f^{-}\) and \(|f|=f^{+}+f^{-}\).
Complete the proof of Theorem \(2,\) in detail.
\(\Rightarrow 4\). Prove Theorem 3.
[Hint: By our conventions, \(A(f \geq g)=A(f-g \geq 0)\) even if \(g\) or \(f\) is \(\pm \infty\) for some \(x \in A .\) (Verify all cases!) By Theorems 1 and \(2, A(f-g \geq 0) \in \mathcal{M} ;\) so \(A(f \geq g) \in \mathcal{M}, \text { and } A(f<g)=A-A(f \geq g) \in \mathcal{M} . \text { Proceed. }]\)
Show that the measurability of \(|f|\) does not imply that of \(f\).
[Hint: Let \(f=1\) on \(Q\) and \(f=-1\) on \(A-Q\) for some \(Q \notin \mathcal{M}(Q \subset A) ;\) e.g., use \(Q\) of Problem \(6 \text { in Chapter } 7, § 8 .]\)
\(\Rightarrow 6\). Show that a function \(f \geq 0\) is measurable on \(A\) iff \(f_{m} \nearrow f\) (pointwise) on \(A\) for some finite simple maps \(f_{m} \geq 0,\left\{f_{m}\right\} \uparrow\).
[Hint: Modify the proof of Lemma \(2,\) setting \(H_{m}=A(f \geq m)\) and \(f_{m}=m\) on \(H_{m}\), and defining the \(A_{m k}\) for \(1 \leq k \leq m 2^{m}\) only.]
\(\Rightarrow 7\). Prove Theorem 3 in \(§ 1 .\)
[Outline: By Problems 7 and 8 in \(\xi 1,\) there are \(q_{i} \in T\) such that
\[
(\forall n) \quad f[A] \subseteq \bigcup_{i=1}^{\infty} G q_{i}\left(\frac{1}{n}\right) .
\]
Set
\[
A_{n i}=A \cap f^{-1}\left[G_{q_{i}}\left(\frac{1}{n}\right)\right] \in \mathcal{M}
\]
by Corollary \(2 ;\) so \(\rho^{\prime}\left(f(x), q_{i}\right)<\frac{1}{n}\) on \(A_{n i}\).
By Corollary 1 in Chapter 7, §1
\[
A=\bigcup_{i=1}^{\infty} A_{n i}=\bigcup_{i=1}^{\infty} B_{n i}(\text {disjoint})
\]
for some sets \(B_{n i} \in \mathcal{M}, B_{n i} \subseteq A_{n i} .\) Now define \(g_{n}=q_{i}\) on \(B_{n i} ;\) so \(\rho^{\prime}\left(f, g_{n}\right)<\frac{1}{n}\) on each \(\left.B_{n i}, \text { hence on } A . \text { By Theorem } 1 \text { in Chapter } 4, §12, g_{n} \rightarrow f \text { (uniformly) on } A .\right]\)
\(\Rightarrow 8\). Prove that \(f: S \rightarrow E^{1}\) is \(\mathcal{M}\) -measurable on \(A\) iff \(A \cap f^{-1}[B] \in \mathcal{M}\) for every Borel set \(B \text { (equivalently, for every open set } B)\) in \(E^{1} .\) (In the case \(f: S \rightarrow E^{*},\) add: "and for \(B=\{\pm \infty\}.\) ")
[Outline: Let
\[
\mathcal{R}=\left\{X \subseteq E^{1} | A \cap f^{-1}[X] \in \mathcal{M}\right\} .
\]
Show that \(\mathcal{R}\) is a \(\sigma\) -ring in \(E^{1}\).
Now, by Theorem \(1,\) if \(f\) is measurable on \(A, \mathcal{R}\) contains all open intervals; for
\[
A \cap f^{-1}[(a, b)]=A(f>a) \cap A(f<b) .
\]
Then by Lemma 2 of Chapter \(7, 2, \mathcal{R} \supseteq \mathcal{G},\) hence \(\mathcal{R} \supseteq \mathcal{B} .\) (Why?)
Conversely, if so,
\[
\left.(a, \infty) \in \mathcal{R} \Rightarrow A \cap f^{-1}[(a, \infty)] \in \mathcal{M} \Rightarrow A(f>a) \in \mathcal{M} .\right]
\]
\(\Rightarrow 9\). Do Problem 8 for \(f: S \rightarrow E^{n}\).
[Hint: If \(f=\left(f_{1}, \ldots, f_{n}\right)\) and \(B=(\overline{a}, \overline{b}) \subset E^{n},\) with \(\bar{a}=\left(a_{1}, \ldots, a_{n}\right)\) and \(\bar{b}=\) \(\left(b_{1}, \ldots, b_{n}\right),\) show that
\[
f^{-1}[B]=\bigcap_{k=1}^{n} f_{k}^{-1}\left[\left(a_{k}, b_{k}\right)\right] .
\]
Apply Problem 8 to each \(f_{k}: S \rightarrow E^{1}\) and use Theorem 2 in §1. Proceed as in Problem \(8 .]\)
Do Problem 8 for \(f: S \rightarrow C^{n},\) treating \(C^{n}\) as \(E^{2 n}\).
Prove that \(f: S \rightarrow\left(T, \rho^{\prime}\right)\) is measurable on \(A\) in \((S, \mathcal{M})\) iff
(i) \(A \cap f^{-1}[G] \in \mathcal{M}\) for every open globe \(G \subseteq T,\) and
(ii) \(f[A]\) is separable in \(T(\text { Problem } 7 \text { in } § 1) .\)
[Hint: If so, proceed as in Problem \(7 \text { (without assuming measurability of } f)\) to show that \(f=\lim g_{n}\) for some elementary maps \(g_{n}\) on \(A .\) For the converse, use Problem 7 in \(§ 1\) and Corollary 2 in \(§ 2\).]
(i) Show that if all of \(T \text { is separable (Problem } 7 \text { in } § 1),\) there is a sequence of globes \(G_{k} \subseteq T\) such that each nonempty open set
\(B \subseteq T\) is the union of some of these \(G_{k}\).
(ii) Show that \(E^{n}\) and \(C^{n}\) are separable.
[Hints: (i) Use the \(G_{q_{i}}\left(\frac{1}{n}\right)\) of Problem 8 in \(§ 1,\) putting them in one sequence.
(ii) Take \(\left.D=R^{n} \subset E^{n} \text { in Problem } 7 \text { of } § 1 .\right]\)
Do Problem 11 with "globe \(G \subseteq T^{\prime \prime}\) replaced by "Borel set \(B \subseteq T\)."
[Hints: Treat \(f\) as \(f: A \rightarrow T^{\prime}, T^{\prime}=f[A],\) noting that
\[
A \cap f^{-1}[B]=A \cap f^{-1}\left[B \cap T^{\prime}\right] .
\]
By Problem \(12,\) if \(B \neq \emptyset\) is open in \(T,\) then \(B \cap T^{\prime}\) is a countable union of "globes" \(G_{q} \cap T^{\prime}\) in \(\left(T^{\prime}, \rho^{\prime}\right) ;\) see Theorem 4 in Chapter \(3, § 12 .\) Proceed as in Problem \(8,\) replacing \(E^{1}\) by \(T\).]
A map \(g:\left(T, \rho^{\prime}\right) \rightarrow\left(U, \rho^{\prime \prime}\right)\) is said to be of Baire class \(0\left(g \in \mathrm{B}_{0}\right)\) on \(D \subseteq T\) iff \(g\) is relatively continuous on \(D .\) Inductively, \(g\) is of Baire class \(n\left(g \in \mathbf{B}_{n}, n \geq 1\right)\) iff \(g=\lim g_{m}\) (pointwise) on \(D\) for some maps \(g_{m} \in \mathbf{B}_{n-1}\). Show by induction that Corollary 4 in \(§ 1\) holds also if \(g \in \mathbf{B}_{n}\) on \(f[A]\) for some \(n .\)