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\title{ HW I for Math 6210\\ due Monday September 04}
\maketitle



\noindent 
1)  Let $X=\mathbb Z$, the ring of integers, and fix a prime $p$. For every $x\in \mathbb Z$ and integer $n\geq 0$, let 
\[
B_{x,n}=\{ y\in \mathbb Z ~|~ y \equiv x \pmod{p^n}\}. 
\] 
Let $\mathcal E \subset \mathcal P(X)$ consist of all disjoint unions of $B_{x,n}$, including the empty set. Prove that $\mathcal E$ is an algebra, and that 
$\mu(B_{x,n})= 1/p^n$ defines a finitely additive measure on it. 

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\noindent 
2) Let $\mathcal A \subseteq \mathcal P(X)$ and $\mathcal B \subseteq \mathcal P(Y)$  be two algebras. Let $\mathcal C \subseteq \mathcal P(X\times Y)$ consist of all disjoint unions of 
products $A\times B$, where $A\in \mathcal A$ and $B\in\mathcal B$. Prove that $\mathcal C$ is an algebra. 

\vskip 10pt 
\noindent 
3) Let $\mathcal A$ be an algebra of sets, with a finitely additive measure $\mu$. 
For any two $A,B\in \mathcal A$ define $d(A,B)=\mu(A\Delta B)$. Prove that $d$ satisfies the triangle inequality i.e. 
\[ 
d(A,C) \leq d(A,B) + d(B,C) 
\] 
for any three sets in $\mathcal A$. 

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\noindent 
4) Let $X=[0,1)\times [0,1)$, and $\mu^*$ the Lebesgue outer measure on $X$. Let $A\subseteq X$. If for every $\epsilon>0$ there exists an elementary set $E$ 
such that $\mu^*(A\Delta E) < \epsilon$, prove that  $\mu^*(A) + \mu^*(A^c)=1$.  

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\noindent 
5) Let $\mu^*$ be an outer measure on $X$. Let $A\subseteq X$ be a measurable set (as defined on page 29 in Folland) 
 $\mu^*(E\cap A) + \mu^*(E\cap A^c)= \mu^*(E)$ for all sets $E$ in $X$. 
If $B$ is a subset of $A^c$ prove that $\mu^*(A\cup B)=\mu^*(A) + \mu^*(B)$.  

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\noindent 
6) Let $\mathcal A$ be a $\sigma$-algebra and $\mu$ a measure on $\mathcal A$. Let $A_1 \supseteq A_2 \supseteq \ldots $ be a sequence of sets 
in $\mathcal A$ such that $\mu(A_1)$ is finite. Let $A=\cap_{i=1}^{\infty} A_i$. Prove that $\lim_{i\mapsto \infty} \mu(A_i)= \mu(A)$. 

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\noindent 
7) Let $\mu$ be a finite Borel measure on $\mathbb R$. Let $F(x)= \mu(-\infty, x]$. Show that $F(x)$ is continuous at $x$ if and only if $\mu(\{x\})=0$. 




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