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\begin{document}

\centerline {\bf Math 523a - Homework 7} 

\bigskip

\begin{enumerate}

\item This is related to the homework problem in the last set 
about conditional expectation. Let $(X,\M,P)$ be measure space
with $P(X)=1$. Such a measure is called a probability measure and 
it is traditional to use capital roman letters like $P$ to 
denote probability measures. 
Let $\{E_n\}^N_{n=1}$ be a collection of disjoint measurable sets 
whose union is $X$.  The collection can be finite (in which case $N$ is 
finite) or countable infinite (in which case $N$ is $\infty$.)
In probability, measurable sets are called events, and 
measurable functions are called random variables. If $f$ is
integrable, then $\int f dP$ is called the expectation or expected 
value of $f$. In probability it is denoted $E[f]$. 
Let $\N$ be the $\sigma$-algebra generated by $\{E_n\}^N_{n=1}$.

\begin{enumerate}
\item Give an explicit characterization of $\N$. In particular, 
how many sets are in $\N$ if $N$ is finite?
\item Given an explicit characterization of the functions that are 
measurable with respect to $\N$. 
\item If $A,B \in \M$ with $P(B)>0$, the conditional probability of $A$
given $B$ is defined to be $P(A|B)=P(A \cap B)/P(B)$. 
As we have seen before, $A \ra P(A|B)$ defines a probability measure. 
If $f$ is integrable, the expectation of $f$ with respect to this 
probability measure is called the conditional expectation of $f$ with 
respect to $B$. We denote if by $E[f|B]$. Show that 
\bea
E[f|B]=\frac{E[1_B f]}{P(B)}
\eea
\item Let $f \in L^1(P)$. In the last homework set you showed that
for any sub $\sigma$-algebra $\N$  there is 
another function $g \in L^1(P)$ which is $\N$ measurable and approximates
$f$ in the sense that 
\bea
\int_E f \, dP  = \int_E g \, dP, \quad   if \, E \in \N
\eea
$g$ is called the conditional expectation of $f$ given $\N$ and 
denoted $E[f|\N]$ in probability. 
Note that $E[f|\N]$ is a function while $E[f|B]$ is a number. 
Give an explicit formula for 
$E[f|\N]$ when $\N$ is the $\sigma$-algebra defined above. 
\end{enumerate}

\item Folland, problem 25, p 100.

\item Folland, problem 26, p 100.

\item Folland, problem 28, p 107.

\item Folland, problem 33, p 108.

\item Folland, problem 36, p 108.

\item Folland, problem 37, p 108.

\end{enumerate}

\end{document}
\end


