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\begin{document}
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\large



\title{Math 533, Spring 2020, HW6} 
\author{Due in class, March 9}
\maketitle

\begin{problem} Let $\mu, \nu$ be Radon measures on $X, Y$, not necessarily $\sigma$-finite.  If $f$ is a nonnegative l.s.c. function on $X \times Y$, show that $x \to \int f_x d\nu$ and $y \to \int f^y d\mu$ are Borel measurable and $\int f d(\mu \hat{\times} \nu) = \iint f d\mu d\nu = \iint f d\nu d\mu$.
\end{problem}


%\begin{problem}
% Prove the following.
% \begin{enumerate}
%  \item $\sB_{X\times Y}^0 \subset \sB_X^0 \otimes \sB_Y^0$.
%  \item If $X$ and $Y$ are either compact or second countable, then $\sB_{X\times Y}^0 = \sB_X^0 \otimes \sB_Y^0$.
%  \item If $X$ is an uncountable set with the discrete topology, then $\sB_{X\times X}^0 \neq \sB_X^0 \otimes \sB_X^0$.
% \end{enumerate}

%\end{problem}

\begin{problem}
The following gives an example of a smooth function not equal to its Taylor expansion at 0.
Let $f(t) = e^{-1/t}$ for $t>0$, $f(t) = 0$ for $t \leq 0$.   Check that
 \begin{enumerate}
  \item For $k \in \bN$ and $t>0$, $f^{(k)}(t) = P_k(1/t) e^{-1/t}$ where $P_k$ is a polynomial.
  \item $f^{(k)}(0)$ exists and is equal to 0 for all $k$.
 \end{enumerate}

\end{problem}

\begin{problem}
 If $f \in L^\infty$ and $\|f^y - f\|_\infty \to 0$ as $y \to 0$, then $f$ agrees a.e. with a uniformly continuous function.
\end{problem}


\end{document}

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