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\title{ HW III for Math 6210\\ due Monday October 09}
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\noindent 
1) Please justify all steps in the following: 
\begin{enumerate} 
\item Prove that $f(x)=x\exp(-\frac{x^2}{2})$ is in $L^1(\mathbb R)$. 
\item Use Theorem 2.27 to compute the derivative of $g(t)=\int_{\mathbb R} \exp(-\frac{x^2}{2}) \cos(tx) dx$, where $t\in \mathbb R$. 
\item Use $\exp(-\frac{x^2}{2})'=-x \exp(-\frac{x^2}{2})$ and integration by parts to show that $g(t)$ satisfies a first order differential equation. 
Solve it to find $g$. 
\end{enumerate} 

\vskip 10pt 
\noindent 
2) Let $\mathcal M \subseteq \mathcal P(X)$ be the $\sigma$ algebra generated by an elementary family $\mathcal E$, see the definition on page 23. 
Let  $\mu$ and $\nu$ be two positive finite measures on $\mathcal M$, such that $\mu(E)=\nu(E)$ for all $E$ in $\mathcal E$. Prove that $\mu=\nu$. 


\vskip 10pt 
\noindent 
In the following exercises $\mu$ denotes the standard Lebesgue measure on the Borel algebra of $\mathbb R^n$.
  Let $P^n$ be the open punctured ball $0< ||x|| <1$ and 
$S^{n-1}$ the sphere $||x||=1$, where $||x||$ is the standard norm on $\mathbb R^n$. 
 Note that $P^n$ is homeomorphic to $(0,1) \times S^{n-1}$. Let $dr$ denote the standard Lebesgue measure on $(0,1)$. 


\vskip 10pt 
\noindent 
3) Take $n=2$ above. Let $d\varphi$ be the Lebesgue measure on the unit circle $S^{1}$ (the arc length).  Let $dr ~d\varphi$ be the product measure on the Borel algebra of 
$(0,1) \times S^1 \cong P^2$. Prove that for every Borel set $E$ in $P^2$, 
\[ 
\mu(E)=\int_E r dr~d\varphi.
\] 

\vskip 10pt 
\noindent 
4) Let $F$ be a Borel set in $S^{n-1}$. Then $c(F)=(0,1)\times F$ is a Borel set in $P^n$. (Why?) Let $\nu(F)=\frac{1}{n} \mu(c(F))$. Prove that $\nu$ is a Borel measure on 
$S^{n-1}$ invariant under the orthogonal transformations. 

\vskip 10pt 
\noindent 
5) Let $\nu$ be the Borel measure on $S^{n-1}$ as in 4). Let $dr \otimes \nu$ denote the product measure on the Borel algebra of 
$(0,1) \times S^{n-1} \cong P^n$. Prove that for every Borel set $E$ in $P^n$, 
\[ 
\mu(E)=\int_E r^{n-1} dr\otimes \nu. 
\] 
Hint: It suffices to do this for $E=(0,a) \times F$, where $F$ is a Borel set in $S^{n-1}$ (why?) and in this case it reduces to proving that $\mu((0,a) \times F)= a^n\mu((0,1) \times F)$. 
Why is the last identity true? 

\vskip 10pt 
\noindent 
The problem 5) combined with 14) on page 52 (the previous HW) gives us the formula for integration in ``polar'' coordinates in $\mathbb R^n$, first on any ball of a finite radius $n$ and 
then by monotone convergence in general. 



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