Measure

Show that if conditions (i) and (iii) in the definition of measure hold, and if $\mu (A)<\infty$ for some $A \in \mathscr {F}$, then condition (ii) holds.

On the $\sigma$-field of all subsets of $\Omega =\left\{ 1,2, \ldots \right\}$ put $\mu (A)=\Sigma_{k \in A^{2}} 2^{-k}$ if $A$ is finite and $\mu (A)=\infty$ otherwise. Is $\mu$ finitely additive? Countably additive?

The natural generalization of (4.9) is

Show that the left-hand inequality always holds. Show that the right-hand inequality holds if $\mu \left(U_{k \geq n} A_{k}\right)<\infty$ for some $n$ but can fail otherwise.

3.10 A measure space $(\Omega , \mathscr {F}, \mu )$ is complete if $A \subset B, B \in \mathscr {F}$, and $\mu (B)=0$ together imply that $A \in \mathscr {F}$-the definition is just as in the probability case. Use the ideas of Problem 3.10 to construct a complete measure space $\left(\Omega , \mathscr {F}^{+}, \mu^{+}\right)$ such that $\mathscr {F} \subset \mathscr {F}^{+}$and $\mu$ and $\mu^{+}$agree on $\mathscr {F}$.

The condition in Theorem 10.2(iv) essentially characterizes $\sigma$-finiteness.

Example 10.5 shows that Theorem 10.3 fails without the $\sigma$-finiteness condition. Construct other examples of this kind.

The proof of Theorem 3.1 obviously applies if the probability measure is replaced by a finite measure, since this is only a matter of rescaling. Take as a starting point then the fact that a finite measure on a field extends uniquely to the generated $\sigma$-field. By the following steps, prove Theorem 11.2-that is, remove the assumption of finiteness.

Suppose that $\mu$ is a nonnegative and finitely additive set function on a semiring $\mathscr {A}$.

Show that Theorem 11.4(ii) can fail if $\mu (B)=\infty$.

This and Problems $11.5,16.12,17.12,17.13$, and 17.14 lead to proofs of the Daniell-Stone and Riesz representation theorems.

Let $\Lambda$ be a real linear functional on a vector lattice $\mathscr {L}$ of (finite) real functions on a space $\Omega$. This means that if $f$ and $g$ lie in $\mathscr {L}$, then so do $f \vee g$ and $f \wedge g$ (with values $\max \left\{ f(\omega ), g(\omega )\right\}$ and $\min \left\{ f(\omega ), g(\omega )\right\}$ ), as well as $\alpha f+\beta g$, and $\Lambda (\alpha f+\beta g)=\alpha \Lambda (f)+\beta \Lambda (g)$. Assume further of $\mathscr {L}$ that $f \in \mathscr {L}$ implies $f \wedge l \in \mathscr {L}$ (where 1 denotes the function identically equal to 1 ). Assume further of $\Lambda$ that it is positive in the sense that $f \geq 0$ (pointwise) implies $\Lambda (f) \geq 0$ and continuous from above at 0 in the sense that $f_{n} \downarrow 0$ (pointwise) implies $\Lambda \left(f_{n}\right) \rightarrow 0$.

Suppose that $\mu$ is a measure on $\mathscr {R}^{1}$ that is finite for bounded sets and is translation-invariant: $\mu (A+x)=\mu (A)$. Show that $\mu (A)=\alpha \lambda (A)$ for some $\alpha \geq 0$. Extend to $R^{k}$

Suppose that $A \in \mathscr {R}^{1}, \lambda (A)>0$, and $0<\theta <1$. Show that there is a bounded open interval $I$ such that $\lambda (A \cap I) \geq \theta \lambda (I)$. Hint: Show that $\lambda (A)$ may be assumed finite, and choose an open $G$ such that $A \subset G$ and $\lambda (A) \geq \theta \lambda (G)$. Now $G=\bigcup_{n} I_{n}$ for disjoint open intervals $I_{n}$ [A12], and $\Sigma_{n} \lambda \left(A \cap I_{n}\right) \geq \theta \sum_{n} \lambda \left(I_{n}\right)$; use an $I_{n}$.

$\uparrow$ If $A \in \mathscr {R}^{1}$ and $\lambda (A)>0$, then the origin is interior to the difference set $D(A)=[x-y: x, y \in A]$. Hint Choose a bounded open interval $I$ as in Problem 12.2 for $\theta =\frac{3}{4}$. Suppose that $\left|z\right|<\lambda (I) / 2$; since $A \cap I$ and $(A \cap I)+z$ are contained in an interval of length less than $3 \lambda (I) / 2$ and hence cannot be disjoint, $z \in D(A)$.

↑ The following construction leads to a subset $H$ of the unit interval that is nonmeasurable in the extreme sense that its inner and outer Lebesgue measures are 0 and $1: \lambda_{*}(H)=0$ and $\lambda^{*}(H)=1$ (see (3.9) and (3.10)). Complete the details. The ideas are those in the construction of a nonmeasurable set at the end of Section 3. It will be convenient to work in $G=[0,1)$; let $\oplus$ and $\Theta$ denote addition and subtraction modulo 1 in $G$, which is a group with identity 0.

$\uparrow$ The construction here gives sets $H_{n}$ such that $H_{n} \uparrow G$ and $\lambda_{*}\left(H_{n}\right)=0$. If $J_{n}=G-H_{n}$, then $J_{n} \downarrow \varnothing$ and $\lambda^{*}\left(J_{n}\right)=1$.

Suppose that $\mu$ is nonnegative and finitely additive on $\mathscr {R}^{k}$ and that $\mu \left(R^{k}\right)<\infty$. Suppose further that $\mu (A)=\sup \mu (K)$, where $K$ ranges over the compact subsets of $A$. Show that $\mu$ is countably additive (Compare Theorem 12.3(ii).)

Suppose $\mu$ is a measure on $\mathscr {R}^{k}$ such that bounded sets have finite measure. Given $A$, show that there exist an $F_{\sigma }$-set $U$ (a countable union of closed sets) and a $G_{\delta }$-set $V$ (a countable intersection of open sets) such that $U \subset A \subset V$ and $\mu (V-U)=0$.

$2.19 \uparrow$ Suppose that $\mu$ is a nonatomic probability measure on $\left(R^{k}, \mathscr {R}^{k}\right)$ and that $\mu (A)>0$. Show that there is an uncountable compact set $K$ such that $K \subset A$ and $\mu (K)=0$.

The minimal closed support of a measure $\mu$ on $\mathscr {R}^{k}$ is a closed set $C_{\mu }$ such that $C_{\mu } \subset C$ for closed $C$ if and only if $C$ supports $\mu$. Prove its existence and uniqueness. Characterize the points of $C_{\mu }$ as those $x$ such that $\mu (U)>0$ for every neighborhood $U$ of $x$. If $k=1$ and if $\mu$ and the function $F(x)$ are related by (12.5), the condition is $F(x-\epsilon )<F(x+\epsilon )$ for all $\epsilon ; x$ is in this case called a point of increase of $F$.

Of minor interest is the $k$-dimensional analogue of (12.4). Let $I_{i}$ be ( $0, t$ ] for $t \geq 0$ and $(t, 0]$ for $t \leq 0$, and let $A_{x}=I_{x_{1}} \times \cdots \times I_{x_{k}}$. Let $\varphi (x)$ be +1 or -1 according as the number of $i, 1 \leq i \leq k$, for which $x_{i}<0$ is even or odd. Show that, if $F(x)=\varphi (x) \mu \left(A_{x}\right)$, then (12.12) holds for bounded rectangles $A$.

Call $F$ degenerate if it is a function of some $k-1$ of the coordinates, the requirement in the case $k=1$ being that $F$ is constant. Show that $\Delta_{A} F=0$ for every bounded rectangle if and only if $F$ is a finite sum of degenerate functions; (12.12) determines $F$ to within addition of a function of this sort.

Let $G$ be a nondecreasing, right-continuous function on the line, and put $F(x, y)=\min \left\{ G(x), y\right\}$. Show that $F$ satisfies the conditions of Theorem 12.5, that the curve $C=\left[(x, G(x)): x \in R^{\prime }\right]$ supports the corresponding measure, and that $\lambda_{2}(C)=0$.

Let $F_{1}$ and $F_{2}$ be nondecreasing, right-continuous functions on the line and put $F\left(x_{1}, x_{2}\right)=F_{1}\left(x_{1}\right) F_{2}\left(x_{2}\right)$. Show that $F$ satisfies the conditions of Theorem 12.5. Let $\mu , \mu_{1}, \mu_{2}$ be the measures corresponding to $F, F_{1}, F_{2}$, and prove that $\mu \left(A_{1} \times A_{2}\right)=\mu_{1}\left(A_{1}\right) \mu_{2}\left(A_{2}\right)$ for intervals $A_{1}$ and $A_{2}$. This $\mu$ is the product of $\mu_{1}$ and $\mu_{2}$; products are studied in a general setting in Section 18.

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 $\lambda U^{-1}$.

The general nondecreasing function $F$ has at most countably many discontinuities. Prove this by considering the open intervals

-each nonempty one contains a rational.

For distribution functions $F$, the second proof of Theorem 14.1 shows how to construct a measure $\mu$ on $(R^{1}, \mathscr {R}^{1})$ such that $\mu (a, b]=F(b)-F(a)$.

↑ Let $C$ be the set of continuity points of $F$.

The Lévy distance $d(F, G)$ between two distribution functions is the infimum of those $\epsilon$ such that $G(x-\epsilon )-\epsilon \leq F(x) \leq G(x+\epsilon )+\epsilon$ for all $x$. Verify that this is a metric on the set of distribution functions. Show that a necessary and sufficient condition for $F_{n} \Rightarrow F$ is that $d\left(F_{n}, F\right) \rightarrow 0$.

12.3 ↑ A Borel function satisfying Cauchy's equation [A20] is automatically bounded in some interval and hence satisfies $f(x)=x f(1)$. Hint: Take $K$ large enough that $\lambda [x: x>s,\left|f(x)\right| \leq K]>0$. Apply Problem 12.3 and conclude that $f$ is bounded in some interval to the right of 0

↑ Consider sets $S$ of reals that are !inearly independent over the field of rationals in the sense that $n_{1} x_{1}+\cdots +n_{k} x_{k}=0$ for distinct points $x_{i}$ in $S$ and integers $n_{i}$ (positive or negative) is impossible unless $n_{i} \equiv 0$.

Show that (14.24) and (14.25) are everywhere infinitely differentiable, although not analytic.