\documentclass[minion]{homework}
\usepackage{cmacros}
\usepackage{graphicx}
\usepackage{color}
\usepackage[all,cmtip]{xy}

\doclabel{Math F641: Homework 13}
\docdate{Due: December 7, 2011}
\begin{document}
\begin{aproblems}
\hproblem 18.11 \solver{David Maxwell}
%\hproblem 18.13 
%\hproblem 18.15
\hproblem 18.17 \solver{TJ Barry}
\hproblem 18.21 \solver{Slava Garayshin}
\hproblem 18.26 \solver{Will Mitchell}
\hproblem 18.36 \solver{Lyman Gilispie}
\hproblem 18.39 \solver{Slava Garayshin}
\hproblem 18.40 \solver{TJ Barry}
\hproblem \solver{Will Mitchell}
Suppose $f:[a,b]\ra [-M,M]$.  Show that $f$ is measurable if and only if
$$
\sup\left\{ \int_a^b \phi : \text{$\phi$ is simple and $\phi\le f$}\right\} =
\inf\left\{ \int_a^b \psi : \text{$\psi$ is simple and $\psi\ge f$}\right\}.
$$	
Conclude that every Riemann integrable function is Lebesgue integrable
and that its Riemann and Lebesgue integrals agree.
\hproblem 18.47 \solver{Lyman Gilispie}
\hproblem 18.55 \solver{David Maxwell}
\hproblem \solver{David Maxwell}
For $t\in\Reals$ and $f\in L_1$, let $f_t(x)=f(x-t)$.  
Show that $f_t\in L_1$ and that the map $t\mapsto f_t$
is continuous from $\Reals$ to $L_1$.

\end{aproblems}
\end{document}

