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\centerline{\bf  Math 240C: Real Analysis, Spring 2020}

\vspace{3 mm}

\centerline{\bf \large  Homework Assignment 1 } 

\vspace{1 mm}

\centerline{\bf Due 12:00 noon, Monday, April 6, 2020}

\bigskip

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%{\bf Name} \rule{2.5 in}{0.005 in}  $ \quad $ {\bf ID number} \rule{2.1 in}{0.005 in}

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\begin{itemize}

\item[1.]
Let $X$ be a locally compact Hausdorff (LCH) space, $Y$ a closed subset of $X$ (which is
an LCH space in the relative topology), and $\mu$ a Radon measure on $Y.$
Define $I: C_{\rm c}(X) \to {\mathbb C}$ by
\[
I(f) = \int_Y (f|_Y) \, d\mu \qquad \forall f \in C_{\rm c}(X),
\]
where $f|_Y$ is the restriction of $f$ onto $Y.$
Prove that $I$ is a positive linear functional on $C_{\rm c}(X)$ and that the induced
Radon measure $\nu$ on $X$ is given by $\nu(E) = \mu(E\cap Y)$ for any 
Borel measurable subset $E$ of $X$. 

\item[2.]
Let $X$ be a locally compact Hausdorff space and $I$ a positive linear functional on $C_{\rm c}(X).$ 
Prove that for any compact subset $K$ of $X$ there exists $ C_K \in \R$, depending on $K$, such that 
$ | I(f) |  \le C_K  \| f \| $ for any $f \in C_{\rm c}(X)$ such that 
$\mbox{supp}\,(f) \subseteq K.$  

%\item[2.]
%Let $\mu$ be a $\sigma$-finite Radon measure on an LCH space $X$ and $A \in {\mathcal B}_X.$
%Prove that the Borel measure $\mu_A$ defined by $\mu_A (E) = \mu(E\cap A)$
%$(E\in {\mathcal B}_X)$ is a Radon measure on $X.$

\item[3.]
Let $\mu$ be a Radon measure on a locally compact Hausdorff space $X$. 
\begin{compactenum}
\item[(1)]
Let $N$ be the union of all open subsets $U \subseteq X$ such that $\mu(U) = 0.$ Prove that
$N$ is the largest open subset of $X$ such that $\mu(N) = 0.$ 
The complement of $N$ is called the {\bf support} of $\mu$
and is denoted by $\mbox{supp}\,(\mu).$
\item[(2)]
Prove that $x \in \mbox{supp}\,(\mu)$ if and only if
\[
\int_X f \, d\mu > 0
\quad \mbox{for all } f \in C_{\rm c}(X,[0,1]) \quad \mbox{such that } f(x) > 0.
\]
\end{compactenum}

\item[4.]
Let $\mu$ be a Radon measure on a locally compact Hausdorff 
space $X$ and $\phi \in L^1(\mu)$ with $\phi \ge 0$ on $X.$ Define
\[
\nu(E) = \int_E \phi  \, d\mu \qquad \forall E\in {\mathcal B}_X.
\]
Prove that $\nu$ is a Radon measure on $X$ and that 
$\mbox{supp}\, (\nu) \subseteq \mbox{supp}\,(\phi) \cap \mbox{supp}\,(\mu)$.

\item[5.]
Let $X$ be a locally compact Hausdorff space and $x_0 \in X.$
Define $I(f) = f(x_0)$ for any $f \in C_{\rm c} (X)$. Prove that 
$I: C_{\rm c} (X) \to \C$ is a positive linear functional on $C_c(X)$ and that 
 the Radon measure associated with the functional $I$ is the Dirac mass $\delta_{x_0}$ at $x_0.$

%\item[6.]
%Prove that the Banach space $C([0,1])$ is not reflexive.

\item[6.]
Let $\mu$ be a Radon measure on a compact Hausdorff space $X$ with $\mu(X) = 1.$
Prove that there is a compact set $K \subseteq X$ such that $\mu(K) = 1$ but
$\mu (H) < 1$ for every proper compact subset $H$ of $K.$

\item[7.] 
Prove that a  Borel measure on $\R^n$ is a Radon measure 
if and only if it is finite on each compact subset of $\R^n .$ 

\end{itemize}

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