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Math 70100: Functions of a Real Variable I \\
Homework 12, due Wednesday, December 10th.\\
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{\bf Name:} Insert your name here.
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We let $L^1$ denote the collection space of integrable functions from $\R \to \R$. 

\begin{questions}
\question {\em (Folland \S2.3 \# 20)} (A generalized Dominated Convergence Theorem) 
Let $f,g \in L^1$ and let $\{f_n\}$ and $\{g_n\}$ be sequences in $L^1$. Show that if $|f_n| \leq g_n$, $f_n \to f$ and $g_n \to g$ pointwise a.e., and $\lim_{n \to \infty} \int g_n = \int g$, then $\int f=\lim_{n \to \infty} \int f_n$. ({\em Hint:} Rework the proof of the dominated convergence theorem, either in class or following Folland. {\em Remark:} The condition that $\int g_n \to \int g$ is necessary as the example $g_n=\frac{1}{n} \chi_{[1,n]}$ illustrates.)
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\question {\em (Folland \S2.3 \# 21)} Suppose $\{f_n \in L^1\}$ is a sequence of functions converging to $f \in L^1$ pointwise a.e., then $\int |f_n-f| \to 0$ if and only if $\int |f_n| \to \int |f|$. ({\em Hint:} Use the prior exercise. {\em Remark:} All the functions need to be integrable here.)
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\question {\em (Folland \S 2.3 \# 19b)} Find a sequence of integrable functions $f_n:\R \to [0,\infty)$ so that $\{f_n\}$ converges uniformly to $f:\R \to [0,\infty)$ but $f$ is not integrable.
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\question {\em (Folland \S 2.3 \# 26)} Show that if $f \in L^1$ and $F(x)=\int_{-\infty}^x f(t)~d \lambda(t)$, then $F:\R \to \R$ is continuous. 
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\question{\em (Pugh Chapter 6 \# 60a)} Let $E \subset \R$ be a measurable set having positive Lebesgue measure. Prove Steinhaus' Theorem: $E$ meets its $t$-translates for all sufficiently small $t \in \R$. ({\em Hint:} density points.)
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\question {\em (Zakeri's Homework 11 \# 4)} Let $f \in L^1$, and let $E$ be a Lebesgue measurable set of positive measure. The average value of $f$ on $E$ is
$$A(f;E)=\frac{1}{\lambda(E)} \int_E f.$$
Prove that if $A(f;E) \in [a,b]$ for every such $E$, then $f(x) \in [a,b]$ for almost every $x$.
\begin{sol}
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\question {\em (Spring 2013 Qual)} Suppose $f:\R \to \R$ is integrable on $[0,1]$, and satisfies $f(x+y)=f(x)f(y)$ for all $x,y \in \R$ with $f(0)=1$. Prove that $f(x)=e^{ax}$ for some constant $a \in \R$. 
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\end{questions}

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