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\begin{document}
\begin{center}
\begin{large}
Math 70100: Functions of a Real Variable I \\
Homework 11, due Wednesday, November 26th.\\
\end{large}
(Email your homework in if necessary!)
\end{center}
{\bf Name:} Insert your name here.
\vspace{1em}
\begin{questions}
\question {\em (Modified from Pugh, Chapter 6 \# 28)}
A non-negative linear combination of measurable characteristic functions is a {\em simple function} (or {\em step function}). That is, a simple function has the form
$$\phi(x)=\sum_{i=1}^n c_i \chi_{E_i}(c)$$
where $E_1, \ldots, E_n$ are measurable sets and $c_1,\ldots, c_n$ are non-negative constants. (The {\em characteristic function} of $E \subset \R$ is the function $\chi_E:\R \to \{0,1\}$ so that $\chi_E(x)=1$ if and only if $x \in E$.)
We say that $\sum c_i \chi_{E_i}$ {\em expresses} $\phi$. If the $E_i$ are disjoint and the $c_i$ are distinct and positive, then the expression for $\phi$ is called {\em canonical}.
\begin{parts}
\part Show that a canonical expression for a simple function exists and is unique. ({\em Remark: It might be useful to review part (b) to see if you want to prove more here.})
\begin{sol}
Insert answer here.
\end{sol}
\part If $\phi$ is a simple function with canonical representation $\sum_{i=1}^n c_i \chi_{E_i}$, define the ``integral'' $I(\phi)=\sum_{i} c_i \lambda(E_i)$. Show that if $\sum_{j=1}^m d_j \chi_{F_j}$ is a (not-necessarily canonical) expression of $\phi$, then
$$I(\phi)=\sum_{j=1}^n d_j \lambda(F_j).$$
\begin{sol}
Insert answer here.
\end{sol}
\part Infer from (b) that the map $I$ from simple functions to $\R$ given by $\phi \mapsto I(\phi)$ is linear.
\begin{sol}
Insert answer here.
\end{sol}
\part Given a measurable function $f:\R \to [0,\infty)$, show there exists
a pointwise increasing sequence of simple functions $\{\phi_n\}$ whose pointwise limit is $f$.
\begin{sol}
Insert answer here.
\end{sol}
\part Show that for any two such sequences $\{\phi_n\}$ and $\{\psi_n\}$ increasing to $f$ as in part (d),
$$\lim_{n \to \infty} I(\phi_n)=\lim_{n \to \infty} I(\psi_n).$$
Therefore, the definition of $I(f)$ as this limit is well-defined.
\part Show that the function $I$ from the space of measurable functions $\R \to [0,\infty)$ to $\R$ given by $f \mapsto I(f)$ is linear.
\begin{sol}
Insert answer here.
\end{sol}
\end{parts}
\question {\em (Pugh, Chapter 6 \# 30)}
Find a sequence of measurable functions $f_n:[0,1] \to [0,1]$ such that
$\int f_n \to 0$ as $n \to \infty$, but for no $x \in [0,1]$ does $f_n(x)$ converge to a limit as $n \to \infty$.
\begin{sol}
Insert answer here.
\end{sol}
\question Let $\{f_n~:~n \in \N\}$ be a sequence of measurable functions $\R \to [0,\infty)$. Define $g_k=\inf_{n \geq k} f_n$, i.e.,
$$g_k(x)=\inf~\{f_n(x)~:~n \geq k\}.$$
Define $h=\liminf_{n \to \infty} f_n$, i.e.,
$$h(x)=\lim_{k \to \infty} g_k(x).$$
\begin{parts}
\part Show that $g_k$ is measurable for all $k \in \N$. Explain why $\int g_k \leq \int f_n$ when $n \geq k$.
\begin{sol}
Insert answer here.
\end{sol}
\part Prove Fatou's lemma. Prove that $h$ is measurable and $\int h \leq \liminf_{n \to \infty} \int f_n$. ({\em Hint: Use the monotone convergence theorem. Remark: Sometimes Fatou's lemma is used to prove the monotone convergence theorem, though we did not do this.})
\begin{sol}
Insert answer here.
\end{sol}
\end{parts}
\question {\em (Pugh, Chapter 6 \# 55)}
A sequence of measurable functions $f_n:[a,b] \to \R$ converges to $f:[a,b] \to \R$ {\em nearly uniformly} if for every $\epsilon$, there is a set $S \subset [a,b]$ with $\lambda(S)<\epsilon$ so that $f_n \to f$ uniformly on $[a,b] \smallsetminus S$.
Show that nearly uniform convergence is transitive in the following sense. Assume $f_n$ converges to $f$ nearly uniformly as $n \to \infty$ and that for each $n$ there is a sequence $f_{n,k}$ which converges nearly uniformly to $f_n$ as $k \to \infty$. (Al functions are measurable and defined on $[a,b]$.)
\begin{parts}
\part Show that there is a sequence $k(n) \to \infty$ such that $f_{n,k(n)}$ converges nearly uniformly to $f$ as $n \to \infty$.
\begin{sol}
Insert answer here.
\end{sol}
\part Why does (a) remain true when almost everywhere convergence replaces nearly uniform convergence?
\begin{sol}
Insert answer here.
\end{sol}
\end{parts}
\end{questions}
\end{document}