\documentclass[11pt,letterpaper,leqno,oneside,fleqn]{book}
\input 600latex.sty
\def\coursename{Statistics 330b/600b, Math 330b spring 2013}
\voffset-0.5in
\topmargin0pt
\textheight 660pt
%\def\ae#1{\text{a.e.}~$[#1]$}
%\def\Leb{{\mathfrak m}}
%\newcommand{\bx}{{\mathbf x}}
%\newcommand{\by}{{\mathbf y}}
\def\Bbar{\overline B}
\def\Bbub{\raise8pt\hbox to 0pt{$\mkern6mu\scriptscriptstyle\circ$\hss}B}
\def\Abub{\bubble{A}}
\def\intC{\bubble{C}}
\def\fbub{\bubble{f}}
\begin{document}
\HW{11}{Thursday 25 April}
{\it Please attempt at least the starred problems.}
\starproblem[BL]
Suppose $f,g\in BL(\xx)$ for some metric space~$\xx$.
\ppart
Define $f^+(x)=\max(f(x),0)$. Show that $\normBL{f^+}\le \normBL{f}$.
\ppart
Show that $\normBL{fg}\le 2\normBL{f}\normBL{g}$.
\ppart
Show that $BL(\xx)$ is a vector space that is stable under pairwise maxima and pairwise products.
\problem[int.ext]
Let $B$ be a subset of a metric space~$\xx$. Show that
\bAlign
\{x: d(x,B) = 0 \} &= \Bbar := \text {closure of } B,\\
\{x:d(x,B^c) > 0 \} &= \Bbub := \text{interior of }B,\\
\{x: d(x,B) = 0 = d(x,B^c)\} &=\Bbar\bsl\Bbub =\partial B
= \text{boundary of }B.
\eAlign
%Hint: If $d(x,B)=0$, there exists points $x_n\in B$ with $d(x,x_n)\to0$.
\problem[smooth] Suppose $K\in\cc^\infty(\RR)$ has compact support, say, $K(z)=0$ for $|z|>1$. Suppose also that~$\ell$ is a bounded (measurable) real function on~$\RR$.
For each $\sig>0$ define
\beqN
\ell_\sig(x) := \int_\RR \ell(x+\sig z) K(z)\,dz
= \sig^{-1}\int_\RR \ell(y) K\left( \frac{y-x}{\sig}\right)\, dy
\eeqN
\ppart
Use a Dominated Convergence argument to show that $\ell_\sig$ has a bounded, continuous derivative.
\ppart
Make repeated appeals to~(i) to deduce that $\ell_\sig\in\cc^\infty(\RR)$, the space of all bounded continuous functions with bounded continuous derivatives of all orders.
\ppart
If $\ell\in BL(\RR)$, show that $\norm{\ell-\ell_\sig}_\infty\to0$ as $\sig\to0$.
\problem[lsc]
Suppose $g$ is a nonnegative function defined on a metric space~$\xx$. Write $\rr$ for the set of all strictly positive rational numbers.
\ppart
Show that $g = \sup_{r\in \rr}r\{g>r\}$, in the pointwise sense.
\ppart
If $g$ is also lower semi-continuous, show that there exists a countable subset~$\ll$ of $BL(\xx)$ whose pointwise supremum equalsf~$g$.
Deduce that $g$ is the pointwise limit of an increasing sequence of functions
in~$BL(\xx)$.
\problem[bub]
Suppose $A$ is a subset of a metric space with interior~$\Abub$. Show that the indicator function of~$\Abub$ equals~$\fbub$, the largest lower semi-continuous function for which $A\ge \fbub$.
\end{document}