• Home
  • Textbooks
  • Probability Theory: A Comprehensive Course
  • Convergence of Measures

Probability Theory: A Comprehensive Course

Achim Klenke

Chapter 13

Convergence of Measures - all with Video Answers

Educators

Section 1

A Topology Primer

02:01

Problem 1

(i) Show that $C([0,1])$ has a separable dense subset.
(ii) Show that the space $\left(C_{b}([0, \infty)),\|\cdot\|_{\infty}\right)$ of bounded continuous functions, equipped with the supremum norm, is not separable.
(iii) Show that the space $C_{c}([0, \infty))$ of continuous functions with compact support, equipped with the supremum norm, is separable.

Runpeng Li
Runpeng Li
Numerade Educator
02:27

Problem 2

Let $\mu$ be a locally finite measure. Show that $\mu(K)<\infty$ for any compact set $K$.

Angelo Rendina
Angelo Rendina
Numerade Educator
01:03

Problem 3

Let $\Omega$ be a Polish space, let $\mu$ be a $\sigma$-finite measure on $(\Omega, \mathcal{B}(\Omega))$ and let $f: \Omega \rightarrow \mathbb{R}$ be a map. Show that the following two statements are equivalent:
(i) There is a Borel measurable map $g: \Omega \rightarrow \mathbb{R}$ with $f=g \mu$-almost everywhere.
(ii) For any $\varepsilon>0$, there is a compact set $K_{\varepsilon}$ with $\mu\left(\Omega \backslash K_{\varepsilon}\right)<\varepsilon$ such that the restricted function $\left.f\right|_{K_{\varepsilon}}$ is continuous.

Carson Merrill
Carson Merrill
Numerade Educator
02:27

Problem 4

Let $\mathcal{U}$ be a family of intervals in $\mathbb{R}$ such that $W:=\bigcup_{U \in \mathcal{U}} U$ has finite Lebesgue measure $\lambda(W) .$ Show that for any $\varepsilon>0$, there exist finitely many pairwise disjoint sets $U_{1}, \ldots, U_{n} \in \mathcal{U}$ with
$$
\sum_{i=1}^{n} \lambda\left(U_{i}\right)>\frac{1-\varepsilon}{3} \lambda(W).
$$
Hint: Choose a finite family $\mathcal{U}^{\prime} \subset \mathcal{U}$ such that $\bigcup_{U \in \mathcal{U}^{\prime}} U$ has Lebesgue measure at least $(1-\varepsilon) \lambda(W)$. Choose a maximal sequence $\mathcal{U}^{\prime \prime}$ (sorted by decreasing lengths) of disjoint intervals and show that each $U \in \mathcal{U}^{\prime}$ is in $(x-3 a, x+3 a)$ for some $(x-a, x+a) \in \mathcal{U}^{\prime \prime}$

Angelo Rendina
Angelo Rendina
Numerade Educator
12:39

Problem 5

Let $C \subset \mathbb{R}^{d}$ be an open, bounded and convex set and assume that $\mathcal{U} \subset\left\{x+r C: x \in \mathbb{R}^{d}, r>0\right\}$ is such that $W:=\bigcup_{U \in \mathcal{U}} U$ has finite Lebesgue measure $\lambda^{d}(W)$. Show that for any $\varepsilon>0$, there exist finitely many pairwise disjoint sets $U_{1}, \ldots, U_{n} \in \mathcal{U}$ such that
$$
\sum_{i=1}^{n} \lambda^{d}\left(U_{i}\right)>\frac{1-\varepsilon}{3^{d}} \lambda(W).
$$
Show by a counterexample that the condition of similarity of the open sets in $\mathcal{U}$ is essential.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:20

Problem 6

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$ and let $A \in \mathcal{B}\left(\mathbb{R}^{d}\right)$ be a $\mu$-null set. Let $C \subset \mathbb{R}^{d}$ be bounded, convex and open with $0 \in C .$ Use Exercise $13.1 .5$ to show that
$$
\lim _{r \downarrow 0} \frac{\mu(x+r C)}{r^{d}}=0 \quad \text { for } \lambda^{d} \text {-almost all } x \in A.
$$
Conclude that if $F$ is the distribution function of a Stieltjes measure $\mu$ on $\mathbb{R}$ and if $A \in \mathcal{B}(\mathbb{R})$ is a $\mu$-null set, then $\frac{d}{d x} F(x)=0$ for $\lambda$-almost all $x \in A$.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
06:22

Problem 7

Let $f \in$ $\mathcal{L}^{1}\left(\mathbb{R}^{d}\right), \mu=f \lambda^{d}$ and let $C \subset \mathbb{R}^{d}$ be open, convex and bounded with $0 \in C$. Show that
$$
\lim _{r \downarrow 0} \frac{\mu(x+r C)}{r^{d} \lambda^{d}(C)}=f(x) \quad \text { for } \lambda^{d}-\text { almost all } x \in \mathbb{R}^{d}.
$$
For the case $d=1$, conclude the fundamental theorem of calculus:
$$
\frac{d}{d x} \int_{[0, x]} f d \lambda=f(x) \quad \text { for } \lambda \text {-almost all } x \in \mathbb{R}.
$$
Hint: Use Exercise 13.1.6 with
$$
\mu_{q}(d x)=(f(x)-q)^{+} \lambda^{d}(d x) \quad \text { for } q \in \mathbb{Q}.
$$
as well as the inequality
$$
\frac{\mu(x+r C)}{r^{d} \lambda^{d}(C)} \leq q+\frac{\mu_{q}(x+r C)}{r^{d} \lambda^{d}(C)}.
$$

Uma Kumari
Uma Kumari
Numerade Educator
01:54

Problem 8

Similarly as in Corollary 13.7, show the following: Let $E$ be a $\sigma$ compact polish space and let $\mu$ be a measure on $E .$ Then $\mu$ is a Radon measure if and only if $\mu(K)<\infty$ for any compact $K \subset E$.

Angelo Rendina
Angelo Rendina
Numerade Educator