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\begin{document}
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\large



\title{Math 533, Spring 2020, HW12} 
\author{Due  May 11}
\maketitle

%\begin{problem}
% Let $\Omega = (\bR^*)^{[0, \infty)}$, $\{X_t\}_{t \geq 0}$ the coordinate functions on $\Omega$, and for any $A \subset [0, \infty)$, $\sM_A = $ the $\sigma$-algebra generated by $\{X_t\}_{t \in A}$.  Prove the following.
% \begin{enumerate}
%  \item If $V \in \sM_A$, $\omega \in V$, and $\omega'(t) = \omega(t)$ for all $t \in A$, then $\omega' \in V$.
%  \item If $V \in \sM_{[0, \infty)}$, then $V \in \sM_A$ for some countable set $A$.
%  \item The space $\Omega_c = C([0,\infty), \bR)$ is not in $\sM_{[0, \infty)}$.
% \end{enumerate}

%\end{problem}


\begin{problem}Prove the following.
\begin{enumerate}
 \item If $f = \sum_1^n c_j \chi_{[a_j, b_j)}$ is a step function, define 
 \[
  I_f(\omega) = \int_0^\infty f(t) d\omega(t) = \sum_1^n c_j \left[\omega(b_j) - \omega(a_j)\right].
 \]
 Then $I_f$ is an $L^2$ random variable on $\Omega_c$ with mean 0 and variance $\|f\|_2^2 = \int |f|^2 dx$.
 \item The map $f \to I_f$ extends to an isometry from $L^2([0, \infty))$ to $L^2(\Omega_c)$.
 \item If $f \in BV([0,\infty))$ is right continuous and $\supp(f)$ is compact, there is a sequence $\{f_n\}$ of step functions such that $f_n \to f$ in $L^2$ and $df_n \to df$ vaguely.
 \item If $f \in BV([0, \infty))$ is right continuous and $\supp(f)$ is compact, then $I_f(\omega) = -\int_0^\infty \omega(t) df(t)$ almost surely.
\end{enumerate}

\end{problem}


\begin{problem}
Let $R_x$ and $L_x$ denote right and left translation by $x$ in a locally compact group $G$. Let $\mu$ be a Radon measure on $G$, and $f \in C_c(G)$.  Show that the functions $x \to \int L_x f d\mu$ and $x \to \int R_x f d\mu$ are continuous.
\end{problem}

\begin{problem}
 Let $G$ be a locally compact group which is homeomorphic to an open subset $U$ of $\bR^n$ in such a way that, if we identify $G$ with $U$, left translation is an affine map -- that is, $xy = A_x(y) + b_x$ where $A_x$ is a linear transformation of $\bR^n$ and $b_x \in \bR^n$.  Show that $|\det A_x|^{-1} dx$ is a left Haar measure on $G$, where $dx$ denotes Lebesgue measure on $\bR^n$.
\end{problem}


\begin{problem}
 Let $G = \begin{pmatrix} x &y\\0&1 \end{pmatrix}$ with $x>0$ and $y \in \bR$.  Construct a Borel set in $G$ with finite left Haar measure but infinite right Haar measure.  Construct a left uniformly continuous function on $G$ that is not right uniformly continuous.
\end{problem}

\begin{problem}
 Let $\{G_\alpha\}_{\alpha \in A}$ be a family of topological groups and $G = \prod_{\alpha \in A} G_\alpha$. Prove that with product topology and coordinatewise multiplication, $G$ is a topological group.  If each $G_\alpha$ is compact and $\mu_\alpha$ is the Haar measure on $G_\alpha$ such that $\mu_\alpha(G_\alpha) = 1$, then the Radon product of the $\mu_\alpha's$ is a Haar measure on $G$.
\end{problem}


\end{document}

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