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\def\semester{Fall 2006}

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{\bf  PROBLEM SET 6 -  Product  measures}
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\begin{enumerate}

\item  Suppose $E \subset  R^2$ is measurable and  that
$E_y =\{ x: (x,y) \in E\} \subset R$ has measure 
zero for almost every $y$. Show that $E$ has measure 
zero.

\item If $f$ and $g$ are measurable on $R$, show that 
     $h(x,y) = f(x) g(y)$ is measurable on $R^2$.

\item  Show 
$$ \int_{R^n} e^{-|x|^2} dm = \pi^{n/2}.$$
Hint:  For $n=1$ use  
 $$ (\int e^{-x^2} dx)^2 = \int \int e^{-x^2-y^2} dxdy,$$
and polar coordinates. For $n>1$, use $|x|^2 = x_1^2 + \dots + x_n^2$
and Fubini's theorem to reduce to the $n=1$ case. 

\item The convolution $f * g$ 
 of two functions $f,g \in L^1 (R) $ is 
defined as the function 
$$ f*g(y) = \int_R  f(y-x)g(x)  dx.$$
Show that $f*g$ is in $L^1(R)$ and  
$$ \int f* g(y) dy = (\int_R f(x) dx)  (\int_R g(x) dx) .$$

\item If $f \in L^1(R)$ define its Fourier transform as 
$$ \hat f(t) = \int_R f(x) e^{-i xt} dx.$$
Show that 
$$ \widehat{f * g} (t) = \hat f(t) \hat g(t).$$
 
\end{enumerate}

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