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\begin{document}

\title{Real Analysis}
\author{Homework 8}
\date{due November 5, 2008}
\maketitle

\begin{enumerate}

\item Let $(\Omega, \Sigma, \mu)$ be a measure space. Let $f \in
  L^p(\Omega)$ for $1 \leq p \leq \infty$.  Show that for every
  $\varepsilon > 0$ there exists a $g \in L^\infty(\Omega)$ such that
\[
  \lVert f-g \rVert_p \leq \varepsilon \,.
\]

\item Show that the finite unions of half-open intervals for the form
  $[a,b)$ where $-\infty \leq a \leq b \leq \infty$ and $a$ and $b$
  are of the form $n \, 2^{-m}$ for some $n \in \mathbb{Z}$, $m \in
  \mathbb{N}$ form an algebra which generates the Borel
  $\sigma$-algebra on $\R$.

\item (From Lieb and Loss.) Suppose the mollifier $j_\varepsilon \in
  C_{\mathrm c}^\infty(\R^n)$ and $f \in C(\R^n)$, show that
  $f_\varepsilon = f * j_\varepsilon$ converges to $f$ pointwise,
  and it does so uniformly on compact subsets of $\R^n$.

\item Let $f_k(x) = \sin(kx)$. Show that $f_k$ converges weakly, but
  not strongly to zero in $L^p([0,1])$ with $1<p<\infty$.
\end{enumerate}

\end{document}
