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\noindent
\Large Math 555\hfill Winter 2015

\noindent
{\bf Homework Set 6\hfill Due Monday, February 23}
\bigskip

\begin{enumerate}
\item
A collection $A_1,A_2,\dots$ of measurable subset of $\R^n$ is
said to be {\it almost disjoint} if 
$$
\l(A_j\cap A_k)=0\qquad\mbox{for}\quad j\ne k\ .
$$
\begin{enumerate}
\item
Prove that if $A_1,A_2,\dots$ are almost disjoint, then
$$\l\left(\bigcup_{k=1}^\infty A_k\right)=
\sum_{k=1}^\infty \l(A_k)\ .
$$
\item
Conversely, suppose that the measurable sets $A_1,A_2,\dots$
satisfy 
$$\l\left(\bigcup_{k=1}^\infty A_k\right)=
\sum_{k=1}^\infty \l(A_k)<\infty\ .
$$
Show that the sets $A_1,A_2,\dots$ are almost disjoint.
Does this remain true if the ``$<\infty$'' is replaced by
the weaker hypothesis that $\l(A_k)<\infty$ for each
$k=1,2,\dots$?
\item
Suppose $A_1,A_2,\dots$ are measurable sets, and suppose that
there is a positive integer $d$ such that
each point $x\in\R^n$ belongs to no more than $d$
of the $A_k$'s. Prove that
$$
\sum_{k=1}^\infty \l(A_k)\le d \l\left(\bigcup_{k=1}^\infty A_k\right)\ .
$$
\end{enumerate}
\item
Define the function on $(0,1)$ as follows:
given $x\in (0,1)$, let $x$ have decimal expansion
$0.a_1a_2a_3\dots$ where each $a_i\in\{0,1,\dots,9\}$
and the expansion is non-repeating whenever possible
(i.e. $0.5=0.49999\dots$) to ensure uniqueness, then set
$$
f(0.a_1a_2a_3\dots)=
\left\{\begin{array}{cl}
\frac{1}{n},&
\mbox{if $n$ is the smallest integer for which $a_n=7$,and}\\
0,&\mbox{otherwise.}
\end{array}\right.
$$
Show that $f$ is measurable. What is $\int_0^1)f(x)dx$?

\noindent
Hint: The Taylor series for $\mbox{ln}(1-x)$ can be useful here.
\item
\begin{enumerate}
\item
Show that the function $\frac{\sin x}{x}$ is not integrable on $(0,\infty)$.
\item
Show that
$\displaystyle{\lim_{R\rightarrow\infty}\int_0^R\frac{\sin x}{x}dx}$ exists.
\item
Show that if $f\ge 0$ is measurable on $(0,\infty)$, then
$$
\lim_{R\rightarrow\infty}\int_0^Rf(x)dx<\infty\quad
\Longleftrightarrow\quad\mbox{$f$ is integrable on $(0,\infty)$.}
$$
\end{enumerate}
%\item
%Evaluate $\int_{-\infty}^\infty e^{-x^2}dx$ by computing
%$$
%\int_\R^2e^{-(x^2+y^2)}dxdy
%$$
%in two ways:
%\begin{enumerate}
%\item
%by directly using Fubini's Theorem, and
%\item
%by using polar coordinates in $\R^2$.
%\end{enumerate}
\item
For the function
$$
f(x,y)=\frac{x-y}{(x+y)^3},
$$
show that the iterated integrals
$$
\int_0^1\left(\int_0^1f(x,y)dy\right)dx\quad\mbox{and}\quad
\int_0^1\left(\int_0^1f(x,y)dx\right)dy
$$
both exist, but are not equal. Show that this does not violate
Fubini's Theorem by showing that
$$
\int_{[0,1]\times[0,1]}|f(x,y)|dxdy=\infty\ .
$$
\item
Let $\map{f}{[0,1]}{\R}$ be a non-negative, continuous, and
strictly increasing function on $[0,1]$. The the inverse function
$f^{-1}$ exists and is continuous on $[f(0),f(1)]$. Define
$$
\cR\int_0^1f(x)dx=\lim_{N\rightarrow\infty}
2^{-N}\sum_{k=1}^{2^N}f(k2^{-N})\qquad\qquad\qquad
\mbox{(the Riemann integral)}
$$
and
$$
\cL\int_0^1f(x)dx=\lim_{N\rightarrow\infty}
\sum_{k=1}^{2^N}y_k\left(f^{-1}(y_k)-f^{-1}(y_{k-1})\right)\qquad
\mbox{(the Lebesgue integral)}
$$
where 
$$
y_k=f(0)+k2^{-N}\left(f(1)-f(0)\right).
$$
\begin{enumerate}
\item
Prove directly that both limits exist.
\item
Prove that
$$
\cR\int_0^1f(x)dx=\cL\int_0^1f(x)dx\ .
$$
For this you may quote a theorem from the theory of Riemann
integration, but give a precise statement and reference.
\end{enumerate}
\end{enumerate}
\end{document}
