Week 2 Worksheet

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1. Prove that every continuous function $f:\mathbb{R}\to \mathbb{R}$ is $(\mathsf{Bor}(\mathbb{R}),\mathsf{Bor}(\mathbb{R}))$ measurable.
2. Let $f_1,f_2,\dots$ be a sequence of $(\mathsf{Bor}(\mathbb{R}),\mathsf{Bor}(\mathbb{R}))$ measurable functions. Suppose that $f_n\to g$ pointwise. Prove that $g$ is $(\mathsf{Bor}(\mathbb{R}),\mathsf{Bor}(\mathbb{R}))$ measurable.
3. Prove that if $f:[a,b]\to \mathbb{R}$ is non-decreasing then $f$ is $(\mathsf{Bor}(\mathbb{R}),\mathsf{Bor}(\mathbb{R}))$ measurable.
4. Fix a function $f:X\to Y$ and a $\sigma$-algebra $\mathcal{C}$ on $Y$. Write down the smallest $\sigma$-algebra $\mathcal{A}$ on $X$ with respect to which $f$ is $(\mathcal{A},\mathcal{C})$ measurable.
5. Given any set $X$ define $$\mathrm{\Sigma }(a)=sup\{a(x_1)+\cdots +a(x_n):\{x_1,\dots ,x_n\}\subset X\}$$ for every $a:X\to [0,\mathrm{\infty }]$.
   1. Prove that $\mathrm{\Sigma }(a+b)=\mathrm{\Sigma }(a)+\mathrm{\Sigma }(b)$.
   2. Take $X=\mathbb{N}\times \mathbb{N}$ and fix $a:X\to [0,\mathrm{\infty })$. Define $b_n(j)=a(n,j)$ for all $n,j\in \mathbb{N}$ and define $c(n)=\mathrm{\Sigma }(b_n)$ for all $n\in \mathbb{N}$. Prove that $\mathrm{\Sigma }(a)=\mathrm{\Sigma }(c)$.
6. Fix a measurable space $(X,\mathcal{B})$.
   1. Prove that if $\mu$ and $\nu$ are measures on $(X,\mathcal{B})$ then so is $t\mu +(1-t)\nu$ for all $0\le t\le 1$.
   2. Given measures ${\mu }_1,{\mu }_2,\dots$ on $(X,\mathcal{B})$ define $$\sum _{n=1}^{\mathrm{\infty }}{\mu }_n$$ and prove it is a measure.
7. Fix a measure space $(X,\mathcal{B},\mu )$. Given $A_1,A_2,\dots$ in $\mathcal{B}$ with $$\sum _{n=1}^{\mathrm{\infty }}\mu (A_n)<\mathrm{\infty }$$ prove that $\mu (\underset{n}{lim sup}{A}_n)=0$.
8. Fix a measure space $(X,\mathcal{B},\mu )$. The symmetric difference of two sets $A,B\subset X$ is the set $$A\mathrm{△}B=A\setminus B\cup B\setminus A=A\mathsf{\text{ xor }}B$$ of points that belong to $A$ or to $B$ but not to $A\cap B$. Define $$\rho :\mathcal{B}\times \mathcal{B}\to [0,\mathrm{\infty }]$$ by $\rho (A,B)=\mu (A\mathrm{△}B)$.
   1. Prove that $\rho$ is symmetric and satisfies the triangle inequality.
   2. Must $\rho$ be a metric on $\mathcal{B}$?
9. Fix an uncountable set $X$ and let $\mathcal{B}$ be the σ-algebra of subsets of $X$ that are either countable or cocountable. Define $$\mu (B)=\{\begin{array}{ll}1& B\mathsf{\text{ cocountable}}\\ 0& B\mathsf{\text{ countable}}\end{array}$$ for all $B\in \mathcal{B}$.
   1. Check that $\mu$ is a measure on $(X,\mathcal{B})$.
   2. Prove that if $\mu (B)>0$ and $A\subset B$ with $\mu (A)<\mu (B)$ then $\mu (A)=0$. A set $B$ with this property is called an atom.
   3. Prove that one cannot write $$\mu =\sum _{n=1}^{\mathrm{\infty }}a(n){\delta }_{x(n)}$$ for points $x(1),x(2),\dots$ in $X$ and positive values $a(1),a(2),\dots$.