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\begin{document}
\begin{center}
{\bf \large Math 702 Problem Set 7} \vs \\
{\bf due Friday 3/28/2014} \vs \vs
\end{center}

\noindent
The symbol $m_k$ denotes Lebesgue measure on $\RR^k$. \vs \vs

\noindent
{\bf Problem 1.} The {\bit graph} and {\bit undergraph} of a measurable function $f: \RR \to [0,+\infty)$ are respectively defined by
$$
\Gamma_f=\{ (x,y) \in \RR^2: y=f(x) \} \quad \text{and} \quad
U_f =\{ (x,y) \in \RR^2: 0\leq y \leq f(x) \}. \vs
$$
\begin{enumerate}
\item[(i)] 
Show that $\Gamma_f$ is measurable and $m_2(\Gamma_f)=0$. \vs
\item[(ii)]
Show that $U_f$ is measurable and $m_2(U_f)=\int_{-\infty}^{\infty} f(x) \, dx$. \vs
\end{enumerate}
(Hint: For measurability issues, look at the function $(x,y) \mapsto f(x)-y$. For computations, use Fubini-Tonelli.) \vs

\noindent
{\bf Problem 2.} Let $f:(0,+\infty) \to (0,+\infty)$ be an orientation-reversing homeomorphism (so $f$ is continuous and strictly decreasing, with $\lim_{x \to 0^+} f(x)=+\infty$ and $\lim_{x \to +\infty} f(x)=0$). Show that 
$$
\int_0^\infty f(x) \, dx = \int_0^{\infty} f^{-1}(x) \, dx. \vs
$$   

\noindent
{\bf Problem 3.} Verify that the function $\sin x/x$ is not Lebesgue integrable on $[0,+\infty)$. On the other hand, use the Fubini-Tonelli theorem and the relation 
$$
\frac{1}{x} = \int_0^{\infty} e^{-xt} \, dt \qquad (x>0)
$$
to show that
$$
\lim_{b \to \infty} \int_0^b \frac{\sin x}{x} \, dx = \frac{\pi}{2}. 
$$ 
This is another illustration of the difference between Lebesgue integral and improper Riemann integral. \vs

\noindent
{\bf Problem 4.} Let $\Sen^{k-1}= \{ u \in \RR^{k} : \| u \| =1 \}$ be the unit sphere in $\RR^{k}$. Identify the product $\Sen^{k-1} \times (0,+\infty)$ with $\RR^k \sm  \{ 0 \}$ by the homeomorphism $\phi: (u,r) \mapsto ru$.
Define a Borel measure $\mu$ on $\Sen^{k-1}$ by 
$$
\mu(E) = k \ m_k (\phi(E \times (0,1))).
$$
Show that for any positive Borel function $f: \RR^k \to \RR$, 
$$
\int_{\RR^k} f \, d\!m_k = \int_0^{\infty} \int_{\Sen^k} r^{k-1} f(ru) \, d\mu \, dr. 
$$
Interpret this formula for $k=2$. \vs

\noindent
{\bf Problem 5.} Recall that the sections of $f: \RR^2 \to \RR$ are the functions $f_x, f^y: \RR \to \RR$ defined by 
$$
f_x(y)=f^y(x)=f(x,y) \qquad \text{for all} \ x,y \in \RR. \vs
$$
\begin{enumerate}
\item[(i)]
If $f_x$ is Borel measurable for all $x$ and $f^y$ is continuous for all $y$, show that $f$ is Borel measurable. \vs
\item[(ii)]
If $f_x$ is Lebesgue measurable for all $x \in \QQ$ and $f^y$ is continuous for a.e. $y$, show that $f$ is Lebesgue measurable. \vs 
\end{enumerate}

\noindent
{\bf Problem 6.} The {\bit Fourier transform} of $f \in L^1(\RR)$ is defined by 
$$
\hat{f}(\omega)= \int_{-\infty}^{\infty} f(x) e^{-i \omega x} \, dx.
$$
\begin{enumerate}
\item[(i)]
Show that $\hat{f}$ is a continuous function on $\RR$ and $\| \hat{f} \|_{\infty} \leq \| f \|_1$. \vs
\item[(ii)]
If $f>0$ everywhere, show that $|\hat{f}(\omega)|<\hat{f}(0)$ for all $\omega \neq 0$. \vs
\item[(iii)]
If $h=f \ast g$ (the convolution), prove that $\hat{h}=\hat{f} \ \hat{g}$. \vs 
\end{enumerate}  

\end{document}

