\[
\newcommand{\C}{\mathbb{C}}
\newcommand{\haar}{\mathsf{m}}
\newcommand{\B}{\mathscr{B}}
\newcommand{\P}{\mathcal{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\g}{>}
\newcommand{\l}{<}
\newcommand{\intd}{\,\mathsf{d}}
\newcommand{\Re}{\mathsf{Re}}
\newcommand{\area}{\mathop{\mathsf{Area}}}
\newcommand{\met}{\mathop{\mathsf{d}}}
\newcommand{\emptyset}{\varnothing}
\DeclareMathOperator{\borel}{\mathsf{Bor}}
\newcommand{\symdiff}{\mathop\triangle}
\DeclareMathOperator{\leb}{\mathsf{Leb}}
\]
Week 2 Worksheet
- Prove that every continuous function $f : \R \to \R$ is $(\borel(\R),\borel(\R))$ measurable.
- Let $f_1,f_2,\dots$ be a sequence of $(\borel(\R),\borel(\R))$ measurable functions. Suppose that $f_n \to g$ pointwise. Prove that $g$ is $(\borel(\R),\borel(\R))$ measurable.
- Prove that if $f : [a,b] \to \R$ is non-decreasing then $f$ is $(\borel(\R),\borel(\R))$ measurable.
- Fix a function $f : X \to Y$ and a $\sigma$-algebra $\mathscr{C}$ on $Y$. Write down the smallest $\sigma$-algebra $\mathscr{A}$ on $X$ with respect to which $f$ is $(\mathscr{A},\mathscr{C})$ measurable.
- Given any set $X$ define
\[
\Sigma(a) = \sup \{ a(x_1) + \cdots + a(x_n) : \{ x_1,\dots,x_n\} \subset X \}
\]
for every $a : X \to [0,\infty]$.
- Prove that $\Sigma(a+b) = \Sigma(a) + \Sigma(b)$.
- Take $X = \N \times \N$ and fix $a : X \to [0,\infty)$. Define $b_n(j) = a(n,j)$ for all $n,j \in \N$ and define $c(n) = \Sigma(b_n)$ for all $n \in \N$. Prove that $\Sigma(a) = \Sigma(c)$.
- Fix a measurable space $(X,\B)$.
- Prove that if $\mu$ and $\nu$ are measures on $(X,\B)$ then so is $t \mu + (1-t) \nu$ for all $0 \le t \le 1$.
- Given measures $\mu_1,\mu_2,\dots$ on $(X,\B)$ define
\[
\sum_{n=1}^\infty \mu_n
\]
and prove it is a measure.
- Fix a measure space $(X,\B,\mu)$. Given $A_1,A_2,\dots$ in $\B$ with
\[
\sum_{n=1}^\infty \mu(A_n) \l \infty
\]
prove that $\mu( \limsup\limits_n A_n ) = 0$.
- Fix a measure space $(X,\B,\mu)$. The symmetric difference of two sets $A,B \subset X$ is the set
\[
A \symdiff B = A \setminus B \cup B \setminus A = A \textsf{ xor } B
\]
of points that belong to $A$ or to $B$ but not to $A \cap B$. Define
\[
\rho : \mathscr{B} \times \mathscr{B} \to [0,\infty]
\]
by $\rho(A,B) = \mu(A \symdiff B)$.
- Prove that $\rho$ is symmetric and satisfies the triangle inequality.
- Must $\rho$ be a metric on $\mathscr{B}$?
- Fix an uncountable set $X$ and let $\B$ be the σ-algebra of subsets of $X$ that are either countable or cocountable. Define
\[
\mu(B) = \begin{cases} 1 & B \textsf{ cocountable} \\
0 & B \textsf{ countable} \end{cases}
\]
for all $B \in \B$.
- Check that $\mu$ is a measure on $(X,\B)$.
- Prove that if $\mu(B) > 0$ and $A \subset B$ with $\mu(A) \l \mu(B)$ then $\mu(A) = 0$. A set $B$ with this property is called an atom.
- Prove that one cannot write
\[
\mu = \sum_{n=1}^\infty a(n) \delta_{x(n)}
\]
for points $x(1),x(2),\dots$ in $X$ and positive values $a(1),a(2),\dots$.