<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt]{article}
\newcommand{\ds}{\displaystyle}
\parindent=0pt
\begin{document}

{\bf Question}

Suppose $f:{\bf R^n}\rightarrow{\bf R^*}$ is measurable and
$m(S)&lt;\infty$.  Suppose also that $A\leq f(x)\leq B$ a.e in $S$.

Show that $f$ is integrable over $S$ and that $\ds Am(S)\leq
\int_S f\leq Bm(S)$


\vspace{0.25in}

{\bf Answer}

Let $g(x)=f(x)$ if $A\leq f(x)\leq B$ but $g(x)=A$ otherwise.

Then $\ds \int_S A=\int AX_S\leq\int gX_S=\int_S g\leq\int
BX_S=\int_S B$

$\ds Am(S)=\int_S A=\int_S g=\int_S f\leq\int_S B=Bm(S)$


\end{document}
</pre></body></html>