Math Problems Vault

Problem F.1

(repeated from Chapter 3) Let $\mu$ be a sigma-finite measure on (T, B). Suppose $h_{1}$ and $h_{2}$ are functions in $\mathcal{M}^{+}(\mathcal{T}, \mathcal{B})$ for which $\mu \left(g(t) h_{1}(t)\right) \leq \mu \left(g(t) h_{2}(t)\right)$ for all $g$ in $\mathcal{M}^{+}(\mathcal{T}, \mathcal{B})$ . Show that $h_{1} \leq h_{2}$ a.e. [ $\mu$ ].

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Problem F.2

Establish Lemma $\langle 3 \rangle$ by the following steps. Write $\gamma$ for the $X$ -marginal of $\Gamma$ .

Lemma 3: Let $\Gamma$ be a finite measure on $\mathcal{A} \otimes \mathcal{B}$ whose $\mathcal{T}$ -marginal is absolutely continuous with respect to the sigma-finite measure $\mu$ on $\mathcal{B}$ . For $f \in \mathcal{M}^{+}(X, \mathcal{A})$ , define the measure $\Gamma_{f}$ on $\mathcal{B}$ by $\Gamma_{f} g: = \Gamma^{x, t}(f(x) g(t))$ for $g \in \mathcal{M}^{+}(\mathcal{T}, \mathcal{B})$ . Then the density $\lambda (t, f): = d \Gamma_{f} / d \mu$ is well defined and determined uniquely up to a $\mu$ -equivalence, and $t \mapsto \lambda (t, f)$ has the following measure-like properties, for all $\alpha_{1}, \alpha_{2}$ in $\mathcal{M}^{+}(\mathcal{T}, \mathcal{B})$ and all $f, f_{1}, f_{2}, \ldots$ in $\mathcal{M}^{+}(X, \mathcal{A})$ : (i) $\lambda \left(t, \alpha_{1} f_{1}+\alpha_{2} f_{2}\right) = \alpha_{1}(t) \lambda \left(t, f_{1}\right)+\alpha_{2}(t) \lambda \left(t, f_{2}\right)$ a.e. $[\mu ]$ ; (ii) if $f_{1} \leq f_{2}$ then $\lambda \left(t, f_{1}\right) \leq \lambda \left(t, f_{2}\right)$ a.e. $[\mu ]$ ; (iii) if $0 \leq f_{1} \leq f_{2} \leq \ldots \uparrow f$ then $\lambda \left(t, f_{n}\right) \uparrow \lambda (t, f)$ a.e. $[\mu ]$ .

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(i)

For fixed $f$ in $\mathcal{M}^{+}(\mathcal{X}, \mathcal{A})$ and each $n$ in $\mathbb {N}$ , write $\lambda_{n}(t, f)$ for any choice of the Radon-Nikodym derivative $d \Gamma_{n \wedge f} / d \mu$ . That is, $\lambda_{n}(t, f)$ is a $\mathcal{B}$ -measurable function for which $\Gamma^{x, t} g(t)(n \wedge f(x)) = \mu^{t} g(t) \lambda_{n}(t, f)$ for all $g$ in $\mathcal{M}^{+}(\mathcal{T}, \mathcal{B})$ . Use Problem [1] to show that $0 \leq \lambda_{n}(t, f) \uparrow$ a.e. [ $\mu$ ].

(ii)

Show that $\lambda (t, f): = \limsup \lambda_{n}(t, f)$ is a density for $\Gamma_{f}$ with respect to $\mu$ . Use Problem [1] to show that the density is unique up to a $\mu$ -equivalence.

(iii)

For fixed $\alpha_{1}, \alpha_{2}, g$ in $\mathcal{M}^{+}(\mathcal{T}, \mathcal{B})$ and $f, f_{1}, f_{2}$ in $\mathcal{M}^{+}(\mathcal{X}, \mathcal{A})$ , show that $\mu^{t}\left(g(t) \alpha_{1}(t) \lambda \left(t, f_{1}\right)\right)+\mu^{t}\left(g(t) \alpha_{2}(t) \lambda \left(t, f_{2}\right)\right) = \Gamma g(t)\left(\alpha_{1}(t) f_{1}(x)+\alpha_{2}(t) f_{2}(x)\right)$ . Deduce via (ii) that $\lambda \left(t, \alpha_{1} f_{1}+\alpha_{2} f_{2}\right) = \alpha_{1}(t) \lambda \left(t, f_{1}\right)+\alpha_{2}(t) \lambda \left(t, f_{2}\right)$ a.e. [ $\mu$ ].

(iv)

If $f_{1}, f_{2} \in \mathcal{M}^{+}(X, \mathcal{A})$ and $f_{1}(x) \leq f_{2}(x)$ a.e. [ $\gamma$ ], show that $\mu^{t}\left(g(t) \lambda \left(t, f_{1}\right)\right) \leq \mu^{t}\left(g(t) \lambda \left(t, f_{1}\right)\right)$ for all $g$ in $\mathcal{M}^{+}(\mathcal{T}, \mathcal{B})$ . Deduce that $\lambda \left(t, f_{1}\right) \leq \lambda \left(t, f_{2}\right)$ a.e. $[\mu ]$ .

(v)

If $f_{1}, f_{2}, \ldots \in \mathcal{M}^{+}(\mathcal{X}, \mathcal{A})$ and $f_{n}(x) \uparrow f(x)$ a.e. [ $\gamma$ ], show that

\mu ^{t}\left(g(t) \lambda \left(t, f_{n}\right)\right) = \Gamma \left(g(t) f_{n}(x)\right) \uparrow \Gamma (g(t) f(x)) = \mu ^{t}(g(t) \lambda (t, f))

for all $g$ in $\mathcal{M}^{+}(\mathcal{T}, \mathcal{B})$ . Deduce that $\lambda \left(t, f_{n}\right) \uparrow \lambda (t, f)$ a.e. $[\mu ]$ .

Problem F.3

Let $\lambda$ be a sigma-finite Radon measure on $(X, \mathcal{A})$ , with $\mathcal{A}$ the Borel sigma-field. Show that there exists a sequence of disjoint compact sets $\left\{ K_{i}: i \in \mathbb {N}\right\}$ for which $\lambda \left(\cup_{i} K_{i}\right)^{c} = 0$ .

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Problem F.4

(i)(ii)(iii)

Let $\lambda$ be a finite measure on the Borel sigma-field of a complete, separable metric space $X$ . Write $\mathcal{K}$ for the class of all compact subsets of $X$ .

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(i)

Show that $\lambda$ is tight: for each $\epsilon > 0$ there exists a $K_{\epsilon }$ in $\mathcal{X}$ such that $\lambda K_{\epsilon }^{c} < \epsilon$ . Hint: For each positive integer $n$ , show that the space $X$ is a countable union of closed balls with radius $1 / n$ . Find a finite family of such balls whose union $B_{n}$ has $\lambda$ measure greater than $\lambda X-\epsilon / 2^{n}$ . Show that $\cap_{n} B_{n}$ is compact, using the total-boundedness characterization of compact subsets of complete metric spaces (Simmons 1963, Section 25).

(ii)

Deduce that $\lambda B = \sup \left\{ \lambda K: B \supseteq K \in \mathcal{X}\right\}$ for each Borel set $B$ . Hint: Compare with the Problems to Chapter 2.

(iii)

Extend the result to sigma-finite measures $\lambda$ on a complete, separable metric space for which $\lambda K < \infty$ for each compact $K$ .

Problem F.5

(i)(ii)(iii)

Let $T$ be an $\mathcal{A} \backslash \mathcal{B}$ -measurable map from $(X, \mathcal{A})$ into ( $y, \mathcal{B}$ ). Suppose $\mathcal{B}$ contains all singleton sets $\left\{ y\right\}$ , for $y \in y$ , and is countably generated: $\mathcal{B}: = \sigma \left(\mathcal{B}_{0}\right)$ for some countable subclass $\mathcal{B}_{0}$ . With no loss of generality you may assume that $\mathcal{B}_{0}$ is stable under complements. Show that $\operatorname {graph}(T) \in \mathcal{A} \otimes \mathcal{B}$ , by following these steps.

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(i)

Say that a set $B$ separates points $y_{1}$ and $y_{2}$ if either $y_{1} \in B$ and $y_{2} \in B^{c}$ or $y_{1} \in B^{c}$ and $y_{2} \in B$ . For fixed $y_{1}$ and $y_{2}$ , show that

\mathcal{B}_{1}: = \left\{ B \in \mathcal{B}: B \text{ does not separate } y_{1} \text{ and } y_{2}\right\}

is a sigma-field. Deduce that $\mathcal{B}_{1}$ cannot contain every $\mathcal{B}_{0}$ set.

(ii)

If $y \neq T x$ , for some $x \in X$ and $y \in y$ , show that $(x, y) \in \left(T^{-1} B^{c}\right) \otimes B$ for some $B \in \mathcal{B}_{0}$ .

(iii)

Deduce that $\operatorname {graph}(T)^{c}$ is a countable union of measurable rectangles.

Problem F.6

Suppose $H: y \rightarrow \mathbb {R}$ is one-to-one. Show that $\sigma (H)$ is countably generated and contains all the singleton sets.

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Problem F.7

(i)(ii)(iii)

Suppose $\mathcal{C}$ is a countably generated sigma-field on a set $y$ , that contains all the singleton subsets. That is, $\mathcal{C}: = \sigma \left(\mathcal{C}_{0}\right)$ , for some countable $\mathcal{C}_{0}: = \left\{ C_{1}, C_{2}, \ldots \right\}$ . Define a real-valued, C -measurable function $H(y): = \sum_{i} 3^{-i}\left\{ y \in C_{i}\right\}$ on $y$ .

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(i)

Show that $H$ is one-to-one. Hint: If $H\left(y_{1}\right) = H\left(y_{2}\right)$ , with $y_{1} \neq y_{2}$ , show that $\mathcal{C}$ cannot separate the points $y_{1}, y_{2}$ , which implies that $\left\{ y_{1}\right\} \notin \mathcal{C}$ .

(ii)

Equip the range $\mathcal{R}: = H^{y}$ (which might not be a Borel set) with the trace sigma-field $\mathcal{D}: = \left\{ B \mathcal{R}: B \in \mathcal{B}(\mathbb {R})\right\}$ . Show that the map $G = H^{-1}$ from $\mathcal{R}$ onto $y$ is $\mathcal{D} \backslash \mathcal{C}$ -measurable. Hint: Show that $G^{-1} C_{i}$ is the set of all points in $\mathcal{R}$ whose base- 3 expansions have a 1 in the $i$ th place.

(iii)

Show that $\mathrm{C} = \sigma (H)$ .

Problem F.8

Suppose $T$ is a function from a set $X$ into a set $\mathcal{T}$ equipped with a sigma-field $\mathcal{B}$ . Suppose $S: X \rightarrow y$ is $\sigma (T) \backslash \mathcal{C}$ -measurable, where $\mathcal{C}$ is a countably generated sigmafield on $y$ containing all the singleton sets. Show that there exists a $\mathcal{B} \backslash \mathcal{C}$ -measurable map $\gamma : \mathcal{T} \rightarrow y$ such that $S = \gamma \circ T$ .

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Problem F.9

(i)(ii)(iii)

Let $X_{0}$ be a countable dense subset of a compact metric space $X$ . Write $\bar{B}(x, r)$ for the closed ball with radius $r$ and center $x$ . Define $\mathcal{X}_{1}$ as the collection of all finite unions of balls $\bar{B}(x, r)$ with $r$ rational and $x \in X_{0}$ . Write $\mathcal{X}_{0}$ for the collection of all finite intersections of sets from $\mathcal{X}_{1}$ .

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(i)

Show that $\mathcal{X}_{0}$ is a countable ( $\emptyset , \cup f, \cap f$ )-paving of compact subsets of $\mathcal{X}$ .

(ii)

For each closed subset $F$ of $X$ show that there exists a decreasing sequence of sets $\left\{ K_{n}\right\}$ from $\mathcal{X}_{0}$ for which $F = \cap_{n} K_{n}$ . Hint: The union of the open balls $B(x, 1 / n)$ with $x \in X_{0} \cap F$ covers the compact $F$ . Find a finite subcover. Consider the union of the corresponding closed balls.

(iii)

For each pair of closed subsets $K_{1}, K_{2}$ in $\mathcal{X}_{0}$ for which $K_{1} \supseteq K_{2}$ show that there exists an increasing sequence of sets $\left\{ H_{n}\right\}$ from $\mathcal{K}_{0}$ for which $U_{n} H_{n} = K_{1} \backslash K_{2}$ . Hint: Cover the compact set $\left\{ x \in K_{1}: d\left(x, K_{2}\right) \geq 2 / n\right\}$ by finitely many closed balls with radius $1 / n$ , then intersect their union with $K_{1}$ .