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\begin{document}
\begin{center}
{\bf \large Math 702 Problem Set 3} \vs \\
{\bf due Friday 2/28/2014} \vs \vs
\end{center}

Unless otherwise stated, $(X,\fM,\mu)$ is a given measure space. \vs 

\noindent
{\bf Problem 1.} Suppose $\mu(X)<+\infty$ and $f_n : X \to \CC$ is a sequence of measurable functions. Show that there is a sequence $\{ c_n \}$ of positive numbers such that $c_n f_n \to 0$ a.e. in $X$. (Hint: For each $n$ there is a large enough $\alpha_n>0$ such that $\mu (\{ x \in X: |f_n(x)| > \alpha_n \}) < 1/2^n$.) \vs

\noindent
{\bf Problem 2.} Let $f \in L^1(\mu)$. Show that for every $\ve>0$ there is a $\delta>0$ such that $\mu(E)<\delta$ implies $\int_E |f| \, d\mu <\ve$. (Hint: Proceed in one of the following ways: (1) Work with the sets $A_n = \{ x \in X: |f(x)|>n \}$ and use the fact that $\int_{A_n} |f| \, d\mu \to 0$ as $n \to \infty$; or (2) Suppose there is an $\ve>0$ and sets $E_n$ with $\mu(E_n)<1/2^n$ such that $\int_{E_n} |f| \, d\mu > \ve$ for all $n$. Look at the integral of $|f|$ over $\bigcup_{k \geq n} E_k$ and let $n \to \infty$ to reach a contradiction.) \vs 

\noindent
{\bf Problem 3.} {\it (Integrability as summability)} Let $f: X \to \CC$ be measurable and define $A_n = \{ x \in X: 2^n < |f(x)| \leq 2^{n+1} \}$ for $n \in \ZZ$. Show that 
$$
f \in L^1(\mu) \quad \text{if and only if} \quad \sum_{n=-\infty}^{\infty} 2^n \mu(A_n) <+\infty. \vs
$$

\noindent
{\bf Problem 4.} Suppose $f_n,f \in L^1(\mu)$, $f_n \to f$ a.e. on $X$, and $\int_X f_n \, d\mu \to \int_X f \, d\mu$. Prove that $\int_E f_n \, d\mu \to \int_E f \, d\mu$ for every measurable set $E \subset X$. (Hint: It suffices to consider the case where $f_n, f$ are positive.) \vs
  
\noindent
{\bf Problem 5.} {\it (Integrals depending on a real parameter)} Let $I$ be an open interval in $\RR$ and $f : X \times I \to \RR$ be a function such that $f(\cdot,t):X \to \RR$ is integrable for each $t \in I$. Define $F: I \to \RR$ by  
$$
F(t)= \int_X f(x,t) \, d\mu. 
$$ 
\begin{enumerate}
\item[(i)]
Suppose $f(x,\cdot):I \to \RR$ is continuous for all $x \in X$ and there is a $g \in L^1(\mu)$ such that $|f(x,t)| \leq g(x)$ for all $(x,t) \in X \times I$. Show that $F$ is continuous on $I$. \vs

\item[(ii)] 
Suppose $f(x,\cdot): I \to \RR$ is differentiable for all $x \in X$ and there is a $g \in L^1(\mu)$ such that $|(\bd f/\bd t)(x,t)| \leq g(x)$ for all $(x,t) \in X \times I$. Show that $F$ is differentiable on $I$ and 
$$
\frac{dF}{dt}= \int_X \frac{\bd f}{\bd t}(x,t) \, d\mu \qquad (t \in I). 
$$
\end{enumerate}
(Hint: For (ii), take any sequence $t_n \to t$ in $I$ and apply the dominated convergence theorem to the sequence of measurable functions $x \mapsto (f(x,t_n)-f(x,t))/(t_n-t)$.) \vs

\noindent
{\bf Problem 6.} Let $\mu(X)<+\infty$. The {\bit average value} of $f \in L^1(\mu)$ over a measurable set $E$ with $\mu(E)>0$ is defined by
$$
f_E = \frac{1}{\mu(E)} \int_E f \, d\mu.
$$
Let $T \subset \CC$ be non-empty and closed. \vs

\begin{enumerate}
\item[(i)]
If $f_E \in T$ for every $E$, show that $f(x) \in T$ for a.e. $x \in X$. \vs

\item[(ii)]
Show by an example that $f(x) \in T$ for a.e. $x \in X$ does not imply $f_E \in T$ for every $E$. \vs

\item[(iii)]
(Bonus) What geometric property of $T$ will guarantee the implication 
$$
f(x) \in T \ \text{for a.e.} \ x \in X \Longrightarrow f_E \in T \ \text{for every} \ E? 
$$
Prove your claim. \vs
\end{enumerate}
(Hint for (i): Express the open set $\CC \sm T$ as a countable union of closed disks $D_n$ and show $f^{-1}(D_n)$ has measure zero for all $n$.) \vs  

\end{document}
