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\begin{document}

\centerline {\bf Math 523a - Homework 4} 

\bigskip

\begin{enumerate}

\item Prove proposition 2.11 on p. 47 of the book. 
{\bf NB:} The definition I gave in class of a.e. was not quite right.
We say that some property which depends on $x$ holds a.e., if there
is a measurable set $E$ with measure zero such that the property
holds for all $x \notin E$. This is not the same as saying the set of 
$x$ for which the property does not hold has measure zero, since the 
set where the property fails may not be a measurable set. 

\item Let $(X,\M)$ be a measurable space. Let $f:X \ra \reals$ 
be a non-negative, measurable function. 
Define a subset of $X \times \reals$ by 
\bea
E=\{ (x,t):f(x) \le t\}
\eea
We use the product $\sigma$-algebra on $X \times \reals$, i.e., 
$\M \otimes \B$ where $\B$ is the Borel $\sigma$-field in $\reals$. 
Prove that $E$ is in $\M \otimes \B$.

\item (Folland, problem 14, p 52). Let $f \ge 0$ be integrable on 
$(X,\M)$. Define $\lambda$ on $\M$ by 
\bea
\lambda(E) = \int_E f \, d \mu, \quad E \in \M
\eea
Prove $\lambda$ is a measure on  $(X,\M)$ and for any integrable non-negative
function $g$, 
\bea
\int g \, d \lambda = \int f \, g \, d \mu
\eea
Hint: First prove it for simple $f$. 

\item (Folland, problem 20, p 59) If $f_n,g_n,f,g$ are non-negative 
integrable functions such that $f_n \ra f$ a.e.,
$g_n \ra g$ a.e., $|f_n| \le g_n$ a.e., and
$\lim_{n \ra \infty} \int g_n \, d\mu = \int g \, d \mu$. Prove that 
$\lim_{n \ra \infty} \int f_n \, d\mu = \int f \, d \mu$.

\item (Folland, problem 21, p 59) Suppose $f_n$ and $f$ are integrable 
and $f_n \ra f$ a.e. Prove $\int |f_n - f| \, d \mu \ra 0$ if and only 
if $\int |f_n| \, d \mu \ra \int |f| \, d \mu$. 

\item Let $m$ be Lebesgue measure on $\reals$. 
Let $f:\reals \ra \reals$ be integrable. Define 
\bea
F(x) = \int_{-\infty}^x f(t) \, dm(t) 
\eea
\begin{enumerate}
\item Prove that $F(x)$ is continuous. 
\item Prove that $F(x)$ is uniformly continuous.
\end{enumerate}

\item (Folland, problem 25, p 59) 
Let $f(x)=x^{-1/2} \chi_{[0,1]}(x)$ on $\reals$. Let $\{r_n\}_{n=1}^\infty$ 
be the rationals in $\reals$. Define 
\bea
g(x) = \sum_{n=1}^\infty 2^{-n} f(x-r_n)
\eea
\begin{enumerate}
\item Prove $g$ is integrable. 
\item Prove $g$ is discontinuous at every point and unbounded on every 
interval, and this is true for any function $h$ such that $g=h$ almost
everywhere. 
\item Prove that $g^2<\infty$ a.e., but $g^2$ is not integrable. 
\end{enumerate}
 
\item (Folland, problem 28 a, p 59)  
Compute the following limit and rigorously justify your computation. 
\bea
\lim_{n \ra \infty} \int_0^\infty \, \left[1+\frac{x}{n}\right]^{-n} 
\sin(\frac{x}{n}) \, dx
\eea

\item For what value of $\alpha$ is the following limit finite and 
non-zero? Compute, with justification, the value of the limit for 
this choice of $\alpha$.
\bea
\lim_{n \rightarrow \infty} n^\alpha \int_0^{\infty} 
\frac{\sin(x)}{(1+x^2)^n} dx.
\eea

\end{enumerate}

\end{document}
\end


