\documentclass[a4paper, 12pt]{amsart}
\usepackage{latexsym, amssymb, amscd, amsfonts, amsthm, amsmath, graphicx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{graphicx}

\newcommand{\sB}{\mathcal{B}}
\newcommand{\sE}{\mathcal{E}}
\newcommand{\sF}{\mathcal{F}}
\newcommand{\sS}{\mathcal{S}}
\newcommand{\sT}{\mathcal{T}}
\newcommand{\sM}{\mathcal{M}}
\newcommand{\sN}{\mathcal{N}}
\newcommand{\sX}{\mathcal{X}}
\newcommand{\sY}{\mathcal{Y}}

\newcommand{\supp}{\operatorname{supp}}

\newcommand{\ux}{\underline{x}}
\newcommand{\uy}{\underline{y}}

\newcommand{\bE}{{\mathbf{E}}}
\newcommand{\zed}{{\mathbb{Z}}}
% \newcommand{\limsup}{{\text{limsup }}}
\newcommand{\cF}{{\mathcal{F}}}
\newcommand{\K}{{\mathcal{K}}}
\newcommand{\cD}{{\mathcal{D}}}
\newcommand{\bS}{{\mathbb S}}
\newcommand{\T}{{\mathcal{T}}}
\newcommand{\G}{{\gamma}}
\newcommand{\A}{{\alpha}}
\newcommand{\Ld}{{\lambda}}
\newcommand{\bZ}{{\mathbb Z}}
\newcommand{\bA}{{\mathbb A}}
\newcommand{\bQ}{{\mathbb Q}}
\newcommand{\bR}{{\mathbb R}}
\newcommand{\bC}{{\mathbb C}}
\newcommand{\bP}{{\mathbb P}}
\newcommand{\bK}{{\bf K}}
\newcommand{\bN}{{\mathbb N}}
\newcommand{\bF}{{\mathbb F}}
\newcommand{\cO}{{\cal O}}
\newcommand{\ic}{{C^{\infty}}}
\newcommand{\X}{{\cal X}}
\newcommand{\pr}{{\prime}}  
\newcommand{\ra}{{\rightarrow}}
\newcommand{\Ra}{{\Rightarrow}}
\newcommand{\la}{{\leftarrow}}
\newcommand{\La}{{\Leftarrow}}
\newcommand{\norm}[1] {\left|\left|#1\right|\right|}
\newcommand{\abs}[1] {\left|#1\right|}
\newcommand{\dlim}[1] {\displaystyle {\lim_{#1}}}
\newcommand{\dsup}[1] {\displaystyle {\sup_{#1}}}
\newcommand{\dinf}[1] {\displaystyle {\inf_{#1}}}
\newcommand{\dmax}[1] {\displaystyle {\max_{#1}}}
\newcommand{\dmin}[1] {\displaystyle {\min_{#1}}}
\newcommand{\dprod}[2]   {\displaystyle {\prod_{#1}^{#2}}}
\newcommand{\dcoprod}[2] {\displaystyle {\coprod_{#1}^{#2}}}
\newcommand{\dsum}[2]    {\displaystyle {\sum_{#1}^{#2}\:}}
\newcommand{\start}[1] {\vskip #1 in\noindent}
\def\postscript#1{\special{#1}}
\newtheorem{defin}{Definition}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{prp}[theorem]{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{example}{Example}
\newtheorem{observation}{Observation}
\theoremstyle{definition}
\newtheorem{problem}{Problem}
\newtheorem*{bonusproblem}{Bonus Problem}

\def\sl{\it}
\def\csc{\it}
\def\focs#1{Proceedings of the #1 IEEE Symposium on the Foundations of
Computer Science}
\def\stoc#1{Proceedings of the #1 ACM Symposium on Theory of Computing}

\def\th{^{th}}
\def\st{^{st}}

\def\bb {\hbox{                 % A box that I use at the end of proofs.
    \vrule height 8pt width .4pt\hskip -.4pt
    \vbox{\hrule width 8pt height .4pt\vskip -.4pt
          \vskip 8pt
          \vskip -.4pt\hrule width 8pt height .4pt}
    \hskip -3.9pt
    \vrule height 8pt width .4pt}\par\medskip}
\def\mapright#1{\buildrel #1 \over \longrightarrow}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapup#1{\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}

\evensidemargin 0in
\oddsidemargin 0in
\textwidth 6.5in
\topmargin 0in
\textheight 8in

\begin{document}
\baselineskip 21pt
\evensidemargin 0.25in
\footnotesep 14pt

\large



\title{Math 533, Spring 2020, HW5} 
\author{Due in class, March 2}
\maketitle

In the following exercises $X$ is a locally compact Hausdorff space.

\begin{problem}
 If $\mu$ is a Radon measure and $f \in L^1(\mu)$, show that $\nu(E) = \int_E f d\mu$ is a Radon measure.
\end{problem}



\begin{problem}
 If $\mu$ is a Radon measure and $f \in L^1(\mu)$ is real-valued, show that for every $\epsilon>0$ there are an l.s.c. function $g$ and a u.s.c. function $h$ such that $h \leq f \leq g$ and $\int (g-h)d\mu < \epsilon$.
\end{problem}



\begin{problem}
 If $\mu$ is a positive Radon measure on $X$ with $\mu(X) = \infty$, show that there exists $f \in C_0(X)$ such that $\int f d\mu = \infty$.  Consequently, every positive linear functional on $C_0(X)$ is bounded.
\end{problem}

\begin{problem}
 If $\mu$ is a $\sigma$-finite Radon measure on $X$ and $\nu \in M(X)$, let $\nu = \nu_1 + \nu_2$ be the Lebesgue decomposition of $\nu$ with respect to $\mu$. Show that $\nu_1$ and $\nu_2$ are Radon.
\end{problem}

\begin{problem} Show that a sequence $\{f_n\}$ in $C_0(X)$ converges weakly to $f \in C_0(X)$ iff $\sup_n \|f_n\|_u < \infty$ and $f_n \to f$ pointwise.
\end{problem}

\begin{problem}
 Find examples of sequences $\{\mu_n\}$ in $M(\bR)$ such that:
 \begin{enumerate}
  \item $\mu_n \to 0$ vaguely, but $\|\mu_n\| \not \to 0$.
  \item $\mu_n \to 0$ vaguely, but $\int f d\mu_n \not \to \int f d\mu$ for some bounded measurable $f$ with compact support.
  \item $\mu_n \geq 0$ and $\mu_n \to \mu$ vaguely, but, there exists $x \in \bR$ such that $F_n(x) \not \to F(x)$.
 \end{enumerate}

\end{problem}


\begin{problem}
 Let $\mu$ be a Radon measure on $X$ such that every open set has positive measure.  Show that for each $x \in X$ there is a sequence $\{f_n\}$ in $L^1(\mu)$ which converges vaguely in $M(X)$ to the point mass at $x$.
\end{problem}



\end{document}

\doc








