\documentclass[12pt]{article}

\usepackage{cite,epic,eepic,euscript,verbatim,amsmath,amssymb,amsthm,afterpage,float,bm,stmaryrd,paralist}

%\usepackage[notcite,notref]{showkeys}
\thispagestyle{empty}
%\pagestyle{plain}

\topmargin -0.3in
\textheight 8.7in
\oddsidemargin -0.1in
\evensidemargin -0in
\textwidth 6.5in

\newcommand{\ve}{{\varepsilon}}

\newcommand{\C}{{\mathbb C}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\Z}{{\mathbb Z}}

\newcommand{\calA}{{\mathcal A}}
\newcommand{\calL}{{\mathcal L}}
\newcommand{\calP}{{\mathcal P}}
\newcommand{\calT}{{\mathcal T}}
\newcommand{\calS}{{\mathcal S}}

\begin{document}

\centerline{\bf  Math 240C: Real Analysis, Spring 2020}

\vspace{3 mm}

\centerline{\bf \large  Homework Assignment 4 } 

\vspace{1 mm}

\centerline{\bf Due 12:00 noon, Monday, April 27, 2020}

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{2 mm}

%{\bf Name} \rule{2.5 in}{0.005 in}  $ \quad $ {\bf ID number} \rule{2.1 in}{0.005 in}

%\vspace{2 mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{itemize}

\item[1.]
Let $X$ and $Y$ be locally compact Hausdorff spaces. Let 
$\mu$ and $\nu$ be Radon measures on $X$ and $Y$, respectively. 
Assume $f \in C_{\rm c} (X \times Y)$. Prove that the functions
\[
x \mapsto \int_Y f_x (y)\, d \nu (y)  \qquad  \mbox{and} \qquad 
y \mapsto \int_X f^y (x)\, d \mu (x)  
\]
are continuous functions on $X$ and $Y$, respectively. 

\item[2.]
Let $X$ and $Y$ be locally compact Hausdorff spaces. Let 
$\mu$ and $\nu$ be Radon measures on $X$ and $Y$, respectively. 
(They are not necessary $\sigma$-finite.)
Assume $f: X \times Y \to \R $ is nonnegative lower semi-continuous. Prove that 
the functions 
\[
x \mapsto \int_Y f_x (y)\, d \nu (y)  \qquad  \mbox{and} \qquad 
y \mapsto \int_X f^y (x)\, d \mu (x)  
\]
are Borel-measurable functions on $X$ and $Y$, respectively, and 
\[
\iint f\, d (\mu \hat{\times} \nu) = \iint f \, d\mu \, d\nu = \iint f \, d\nu \, d \mu. 
\]

\item[3.]
Let $p = p(x)$ $(x \in \R^n)$ be a polynomial and define $f(x) = p(x) e^{-|x|^2}$
$(x\in \R^n).$ Prove that $f \in \calS.$

\item[4.]
Let $\eta_t = e^{-1/t} $ for $t > 0$ and $\eta (t) = 0$ for $t \le 0.$
Prove the following:
\begin{compactenum}
\item[(1)]
If $k \in \N$ and $t > 0$ then $\eta^{(k)} (t) = P_k(1/t) e^{-1/t} $, where 
$P_k$ is a polynomial of degree $2k;$
\item[(2)]
$\eta^{(k)}(0)$ exists and equals zero for all $k \in \N.$
\end{compactenum}

\item[5.]
If $f \in L^\infty(\R^n)$ be such that $\| \tau_y f - f \|_\infty \to 0$ as $y \to 0$. Prove 
that $f $ agrees a.e.\ with a uniformly continuous function. (See the hint for 
Exercise 4 on page 239.)

\item[6.]
Let $f \in L^1_{\rm loc}(\R^n)$  and $g\in C^k_{\rm c}(\R^n)$. Prove that $f*g \in C^k (\R^n)$
and $\partial^\alpha ( f * g ) = f * \partial^\alpha g$ for all $\alpha$ with $| \alpha | \le k.$ 

\end{itemize}

\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

