A modicum of measure theory

Suppose events $A_{1}, A_{2}, \ldots$, in a probability space $(\Omega , \mathcal{F}, \mathbb {P})$, are independent: meaning that $\mathbb {P}\left(A_{i_{1}} A_{i_{2}} \ldots A_{i_{k}}\right) = \mathbb {P}\left(A_{i_{1}}\right) \mathbb {P}\left(A_{i_{2}} \ldots \right) \mathbb {P}\left(A_{i_{k}}\right)$ for all choices of distinct subscripts $i_{1}, i_{2}, \ldots , i_{k}$, all $k$. Suppose $\sum_{i = 1}^{\infty } \mathbb {P}\left(A_{i}\right) = \infty$.

Let $A_{1}, A_{2}, \ldots$ be events in a probability space $(\Omega , \mathcal{F}, \mathbb {P})$. Define $X_{n} = A_{1}+\ldots +A_{n}$ and $\sigma_{n} = \mathbb {E}\left[X_{n}\right]$. Suppose $\sigma_{n} \rightarrow \infty$ and $\left\| X_{n} / \sigma_{n}\right\|_{2} \rightarrow 1$. (Compare with the inequality $\left\| X_{n} / \sigma_{n}\right\|_{2} \geq 1$, which follows from Jensen's inequality.)

Suppose $T$ is a function from a set $X$ into a set $\mathcal{Y}$, and suppose that $\mathcal{Y}$ is equipped with a $\sigma$-field $\mathcal{B}$. Define $\mathcal{A}$ as the sigma-field of sets of the form $T^{-1} B$, with $B$ in $\mathcal{B}$. Suppose $f \in \mathcal{M}^{+}(X, \mathcal{A})$. Show that there exists a $\mathcal{B} \backslash \mathcal{B}[0, \infty ]$-measurable function $g$ from $\mathcal{Y}$ into $[0, \infty ]$ such that $f(x) = g(T(x))$, for all $x$ in $X$, by following these steps.

Let $g_{1}, g_{2}, \ldots$ be $\mathcal{A} \backslash \mathcal{B}(\mathbb {R})$-measurable functions from $X$ into $\mathbb {R}$. Show that $\left\{ \lim \sup_{n} g_{n} > t\right\} = \bigcup_{\substack {r \in \mathbb {Q} \\ r > t}} \bigcap_{m = 1}^{\infty } \bigcup_{i \geq m}\left\{ g_{i} > r\right\}$. Deduce, without any appeal to Example $\langle 8 \rangle$, that $\lim \sup g_{n}$ is $\mathcal{A} \backslash \mathcal{B}(\overline{\mathbb {R}})$-measurable. Warning: Be careful about strict inequalities that turn into nonstrict inequalities in the limit-it is possible to have $x_{n} > x$ for all $n$ and still have $\lim \sup_{n} x_{n} = x$.

Suppose a class of sets $\mathcal{E}$ cannot separate a particular pair of points $x, y$ : for every $E$ in $\mathcal{E}$, either $\left\{ x, y\right\} \subseteq E$ or $\left\{ x, y\right\} \subseteq E^{c}$. Show that $\sigma (\mathcal{E})$ also cannot separate the pair.

A collection of sets $\mathcal{F}_{0}$ that is stable under finite unions, finite intersections, and complements is called a field. A nonnegative set function $\mu$ defined on $\mathcal{F}_{0}$ is called a finitely additive measure if $\mu \left(\cup_{i \leq n} F_{i}\right) = \sum_{i \leq n} \mu F_{i}$ for every finite collection of disjoint sets in $\mathcal{F}_{0}$. The set function is said to be countably additive on $\mathcal{F}_{0}$ if $\mu \left(\cup_{i \in \mathbb {N}} F_{i}\right) = \sum_{i \in \mathbb {N}} \mu F_{i}$ for every countable collection of disjoint sets in $\mathcal{F}_{0}$ whose union belongs to $\mathcal{F}$. Suppose $\mu X < \infty$. Show that $\mu$ is countably additive on $\mathcal{F}_{0}$ if and only if $\mu A_{n} \downarrow 0$ for every decreasing sequence in $\mathcal{F}_{0}$ with empty intersection.

Let $f_{1}, \ldots , f_{n}$ be functions in $\mathcal{M}^{+}(X, \mathcal{A})$, and let $\mu$ be a measure on $\mathcal{A}$. Show that $\mu \left(\vee_{i} f_{i}\right) \leq \sum_{i} \mu f_{i} \leq \mu \left(\vee_{i} f_{i}\right)+\sum_{i < j} \mu \left(f_{i} \wedge f_{j}\right)$ where $\vee$ denotes pointwise maxima of functions and $\wedge$ denotes pointwise minima.

Let $\mu$ be a finite measure and $f$ be a measurable function. For each positive integer $k$, show that $\mu \left|f\right|^{k} < \infty$ if and only if $\sum_{n = 1}^{\infty } n^{k-1} \mu \left\{ \left|f\right| \geq n\right\} < \infty$.

Suppose $v: = T \mu$, the image of the measure $\mu$ under the measurable map $T$. Show that $f \in \mathcal{L}^{1}(\nu )$ if and only if $f \circ T \in \mathcal{L}^{1}(\mu )$, in which case $v f = \mu (f \circ T)$.

Let $\left\{ h_{n}\right\} ,\left\{ f_{n}\right\}$, and $\left\{ g_{n}\right\}$ be sequences of $\mu$-integrable functions that converge $\mu$ almost everywhere to limits $h, f$ and $g$. Suppose $h_{n}(x) \leq f_{n}(x) \leq g_{n}(x)$ for all $x$. Suppose also that $\mu h_{n} \rightarrow \mu h$ and $\mu g_{n} \rightarrow \mu g$. Adapt the proof of Dominated Convergence to prove that $\mu f_{n} \rightarrow \mu f$.

A collection of sets is called a monotone class if it is stable under unions of increasing sequences and intersections of decreasing sequences. Adapt the argument from Theorem $\langle 38 \rangle$ to prove: if a class $\mathcal{E}$ is stable under finite unions and complements then $\sigma (\mathcal{E})$ equals the smallest monotone class containing $\mathcal{E}$.

Let $\mu$ be a finite measure on the Borel sigma-field $\mathcal{B}(X)$ of a metric space $X$. Call a set $B$ inner regular if $\mu B = \sup \left\{ \mu F: B \supseteq F \text{ closed}\right\}$ and outer regular if $\mu B = \inf \left\{ \mu F: B \subseteq G \text{ open}\right\}$

Let $\mu$ be a finite measure on the Borel sigma-field of a complete, separable metric space $X$. Show that $\mu$ is tight: for each $\epsilon > 0$ there exists a compact $K_{\epsilon }$ such that $\mu K_{\epsilon }^{c} < \epsilon$.

A sequence of random variables $\left\{ X_{n}\right\}$ is said to converge in probability to a random variable $X$, written $X_{n} \xrightarrow {\mathbb {P}} X$, if $\mathbb {P}\left(\left|X_{n}-X\right| > \epsilon \right) \rightarrow 0$ for each $\epsilon > 0$.

Let $f$ and $g$ be measurable functions on $(X, \mathcal{A}, \mu )$, and $r$ and $s$ be positive real numbers for which $r^{-1}+s^{-1} = 1$. Show that $\mu \left|f g\right| \leq \left(\mu \left|f\right|^{r}\right)^{1 / r}\left(\mu \left|g\right|^{s}\right)^{1 / s}$ by arguing as follows. First dispose of the trivial case where one of the factors on the righthand side is 0 or $\infty$. Then, without loss of generality (why?), assume that $\mu \left|f\right|^{r} = 1 = \mu \left|g\right|^{s}$. Use concavity of the logarithm function to show that $\left|f g\right| \leq \left|f\right|^{r} / r+\left|g\right|^{s} / s$, and then integrate with respect to $\mu$. This result is called the Hölder inequality.

Generalize the Hölder inequality (Problem [15]) to more than two measurable functions $f_{1}, \ldots , f_{k}$, and positive real numbers $r_{1}, \ldots , r_{k}$ for which $\sum_{i} r_{i}^{-1} = 1$. Show that $\mu \left|f_{1} \ldots f_{k}\right| \leq \prod_{i}\left(\mu \left|f_{i}\right|^{r_{i}}\right)^{1 / r_{i}}$.

Let $(X, \mathcal{A}, \mu )$ be a measure space, $f$ and $g$ be measurable functions, and $r$ be a real number with $r \geq 1$. Define $\left\| f\right\|_{r} = \left(\mu \left|f\right|^{r}\right)^{1 / r}$. Follow these steps to prove Minkowski's inequality: $\left\| f+g\right\|_{r} \leq \left\| f\right\|_{r}+\left\| g\right\|_{r}$.

For $f$ in $\mathcal{L}^{1}(\mu )$ define $\left\| f\right\|_{1} = \mu \left|f\right|$. Let $\left\{ f_{n}\right\}$ be a Cauchy sequence in $\mathcal{L}^{1}(\mu )$, that is, $\left\| f_{n}-f_{m}\right\|_{1} \rightarrow 0$ as $\min (m, n) \rightarrow \infty$. Show that there exists an $f$ in $\mathcal{L}^{1}(\mu )$ for which $\left\| f_{n}-f\right\|_{1} \rightarrow 0$, by following these steps.

Let $\left\{ f_{n}\right\}$ be a Cauchy sequence in $\mathcal{L}^{p}(X, \mathcal{A}, \mu )$, that is, $\left\| f_{n}-f_{m}\right\|_{p} \rightarrow 0$ as $\min (m, n) \rightarrow \infty$. Show that there exists a function $f$ in $\mathcal{L}^{p}(X, \mathcal{A}, \mu )$ for which $\left\| f_{n}-f\right\|_{p} \rightarrow 0$, by following these steps.

For each random variable on a probability space $(\Omega , \mathcal{F}, \mathbb {P})$ define

Let $L^{\infty }: = L^{\infty }(\Omega , \mathcal{F}, \mathbb {P})$ denote the set of equivalence classes of real-valued random variables with $\left\| X\right\|_{\infty } < \infty$. Show that $\left\| \cdot \right\|_{\infty }$ is a norm on $L^{\infty }$, which is a vector space, complete under the metric defined by $\left\| X\right\|_{\infty }$.

Let $\left\{ X_{t}: t \in T\right\}$ be a collection of $\overline{\mathbb {R}}$-valued random variables with possibly uncountable index set $T$. Complete the following argument to show that there exists a countable subset $T_{0}$ of $T$ such that the random variable $X = \sup_{t \in T_{0}} X_{t}$ has the properties (a) $X \geq X_{t}$ almost surely, for each $t \in T$ (b) if $Y \geq X$, almost surely, for each $t \in T$, then $Y \geq X$ almost surely (The random variable $X$ is called the essential supremum of the family. It is denoted by ess $\sup_{t \in T} X_{t}$. Part (b) shows that it is, unique up to an almost sure equivalence.)

Let $\Psi$ be a convex, increasing function for which $\Psi (0) = 0$ and $\Psi (x) \rightarrow \infty$ as $x \rightarrow \infty$. (For example, $\Psi (x)$ could equal $x^{p}$ for some fixed $p \geq 1$, or $\exp (x)-1$ or $\exp \left(x^{2}\right)-1$.) Define $\mathcal{L}^{\Psi }(X, \mathcal{A}, \mu )$ to be the set of all real-valued measurable functions on $X$ for which $\mu \Psi \left(\left|f\right| / c_{0}\right) < \infty$ for some positive real $c_{0}$. Define $\left\| f\right\|_{\Psi }: = \inf \left\{ c > 0: \mu \Psi (\left|f\right| / c) \leq 1\right\}$, with the convention that the infimum of an empty set equals $+\infty$. For each $f, g$ in $\mathcal{L}^{\Psi }(X, \mathcal{A}, \mu )$ and each real $t$ prove the following assertions.

Define $\left\| f\right\|_{\Psi }$ and $\mathcal{L}^{\Psi }$ as in Problem [22]. Let $\left\{ f_{n}\right\}$ be a Cauchy sequence in $\mathcal{L}^{\Psi }(\mu )$, that is, $\left\| f_{n}-f_{m}\right\|_{\Psi } \rightarrow 0$ as $\min (m, n) \rightarrow \infty$. Show that there exists an $f$ in $\mathcal{L}^{\Psi }(\mu )$ for which $\left\| f_{n}-f\right\|_{\Psi } \rightarrow 0$, by following these steps.

Let $\Psi$ be a convex increasing function with $\Psi (0) = 0$, as in Problem [22]. Let $\Psi^{-1}$ denote its inverse function. If $X_{1}, \ldots , X_{N} \in \mathcal{L}^{\Psi }(X, \mathcal{A}, \mu )$, show that

For each $\theta$ in [0, 1] let $X_{n, \theta }$ be a random variable with a $\operatorname {Binomial}(n, \theta )$ distribution. That is, $\mathbb {P}\left(X_{n, \theta } = k\right) = \binom {n}{k} \theta^{k}(1-\theta )^{n-k}$ for $k = 0,1, \ldots , n$. You may assume these elementary facts: $\mathbb {E}\left[X_{n, \theta }\right] = n \theta$ and $\mathbb {E}\left[\left(X_{n, \theta }-n \theta \right)^{2}\right] = n \theta (1-\theta )$. Let $f$ be a continuous function defined on $[0,1]$.

Extend the approximation result from Example $\langle 46 \rangle$ to the case of an infinite measure $\mu$ on $\mathcal{B}\left(\mathbb {R}^{k}\right)$ that gives finite measure to each compact set.