Measurable Functions and Mappings

Functions are often defined in pieces (foi example, let $f(x)$ be $x^{3}$ or $x^{-1}$ as $x \geq 0$ or $x<0$ ), and the following result shows that the function is measurable if the pieces are.

Consider measurable spaces $(\Omega , \mathscr {F})$ and $(\Omega^{\prime }, \mathscr {F}^{\prime })$ and a map $T \quad \Omega \rightarrow \Omega^{\prime }$ Let $A_{1}, A_{2}, \ldots$ be a countable covering of $\Omega$ by $\mathscr {F}$-sets. Consider the $\sigma$-field $\mathscr {F}_{n}=\left[A: A \subset A_{n}, A \in \mathscr {F}\right]$ in $A_{n}$ and the restriction $T_{n}$ of $T$ to $A_{n}$. Show that $T$ is measurable $\mathscr {F} / \mathscr {F}^{\prime }$ if and only if $T_{n}$ is measurable $\mathscr {F}_{n} / \mathscr {F}^{\prime }$ for each $n$.

$\uparrow$ Suppose that $f: \Omega \rightarrow R^{1}$. Show that $f$ is measurable $T^{-1} \mathscr {F}^{\prime }$ if and only if there exists a map $\varphi : \Omega^{\prime } \rightarrow R^{1}$ such that $\varphi$ is measurable $\mathscr {F}^{\prime }$ and $f=\varphi T$. Hint: First consider simple functions and then use Theorem 13.5.

↑ Relate the result in Problem 13.3 to Theorem 5.1(ii).

Show of real functions $f$ and $g$ that $f(\omega )+g(\omega )<x$ if and only if there exist rationals $r$ and $s$ such that $r+s<x, f(\omega )<r$, and $g(\omega )<s$. Prove directly that $f+g$ is measurable $\mathscr {F}$ if $f$ and $g$ are.

Let $\mathscr {F}$ be a $\sigma$-field in $R^{1}$. Show that $\mathscr {R}^{1} \subset \mathscr {F}$ if and only if every continuous function is measurable $\mathscr {F}$. Thus $\mathscr {R}^{1}$ is the smallest $\sigma$-field with respect to which all the continuous functions are measurable.

Consider on $R^{!}$the smallest class $\mathscr {X}$ (that is, the intersection of all classes) of real functions containing all the continuous functions and closed under pointwise passages to the limit. The elements of $\mathscr {X}$ are called Baire functions. Show that Baire functions and Borel functions on $R^{1}$ are the same thing.

A real function $f$ on the line is upper semicontinuous at $x$ if for each $\epsilon$ there is a $\delta$ such that $\left|x-y\right|<\delta$ implies that $f(y)<f(x)+\epsilon$. Show that, if $f$ is everywhere upper semicontinuous, then it is measurable.

${ }^{\dagger }$ But see Problem 13.14.

Suppose that $f_{n}$ and $f$ are finite-valued, $\mathscr {F}$-measurable functions such that $f_{n}(\omega ) \rightarrow f(\omega )$ for $\omega \in A$, where $\mu (A)<\infty$ ( $\mu$ a measure on $\mathscr {F}$ ). Prove Egoroff's theorem: For each $\epsilon$ there exists a subset $B$ of $A$ such that $\mu (B)<\epsilon$ and $f_{n}(\omega ) \rightarrow f(\omega )$ uniformly on $A-B$. Hint: Let $B_{n}^{(k)}$ be the set of $\omega$ in $A$ such that $\left|f(\omega )-f_{i}(\omega )\right|>k^{-1}$ for some $i \geq n$. Show that $B_{n}^{(k)} \downarrow / \varnothing$ as $n \uparrow \infty$, choose $n_{k}$ so that $\mu \left(B_{n_{k}}^{(k)}\right)<\epsilon / 2^{k}$, and put $B=\bigcup_{k=1}^{\infty } B_{n_{k}}^{(k)}$.

$\uparrow$ Show that Egoroff's theorem is false without the hypothesis $\mu (A)<\infty$.

$2.9 \uparrow$ Show that, if $f$ is measurable $\sigma (\mathscr {A})$, then there exists a countable subclass $\mathscr {A}_{f}$ of $\mathscr {A}$ such that $f$ is measurable $\sigma \left(\mathscr {A}_{f}\right)$.

Circular Lebesgue measure Let $C$ be the unit circle in the complex plane, and define $T:[0,1) \rightarrow \mathcal{C}$ by $T \omega =e^{2 \pi i \omega }$. Let $\mathscr {B}$ consist of the Borel subsets of $[0, i)$, and let $\lambda$ be Lebesgue measure on $\mathscr {B}$. Show that $\mathscr {b}=\left[A: T^{-1} A \in \mathscr {B}\right]$ consists of the sets in $\mathscr {R}^{2}$ (identify $R^{2}$ with the complex plane) that are contained in $C$ Show that $\mathscr {C}$ is generated by the arcs of $C$. Circular Lebesgue measure is defined as $\mu =\lambda T^{-1}$. Show that $\mu$ is invariant under rotations: $\mu [\theta z: z \in A]= \mu (A)$ for $A \in \mathscr {C}$ and $\theta \in C$.

$\uparrow$ Suppose that the circular Lebesgue measure of $A$ satisfies $\mu (A)>1-n^{-1}$ and that $B$ contains at most $n$ points. Show that some rotation carries $B$ into $A \theta B \subset A$ fol some $\theta$ in $C$.

Show by example that $\mu \sigma$-finite does not imply $\mu T^{-1} \sigma$-finite.

Consider Lebesgue measure $\lambda$ restricted to the class $\mathscr {B}$ of Borel sets in ( 0,1 ]. For a fixed permutation $n_{1}, n_{2}, \ldots$ of the positive integers, if $x$ has dyadic expansion.$x_{1} x_{2} \ldots$, take $T x=. x_{n_{1}} x_{n_{2}} \ldots$. Show that $T$ is measurable $\mathscr {B} / \mathscr {B}$ and that $\lambda T^{-1}=\lambda$.

Let $H_{k}$ be the union of the intervals $\left((i-1) / 2^{k}, i / 2^{k}\right]$ for $i$ even, $1 \leq i \leq 2^{k}$. Show that if $0<f(\omega ) \leq 1$ for all $\omega$ and $A_{k}=f^{-1}\left(H_{k}\right)$, then $f(\omega )= \sum_{k=1}^{\infty } I_{A_{k}}(\omega ) / 2^{k}$, an infinite linear combination of indicators.

Let $S=\left\{ 0,1\right\}$, and define a map $T$ from sequence space $S^{\infty }$ to $[0,1]$ by $T \omega =\sum_{k=1}^{\infty } a_{k}(\omega ) / 2^{k}$. Define a map $U$ of $[0,1]$ to $S^{\infty }$ by $U x=\left(d_{1}(x), d_{2}(x), \ldots \right)$, where the $d_{k}(x)$ are the digits of the nonterminating dyadic expansion of $x$ (and $d_{k}(0) \equiv 0$ ). Show that $T$ is measurable $\mathscr {C} / \mathscr {B}$ and that $U$ is measurable $\mathscr {B} / \mathscr {C}$. Let $P$ be the measure specified by (2.21) for $p_{0}=p_{1}=\frac{1}{2}$. Describe $P T^{-1}$ and $\lambda U^{-1}$.