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\begin{document}

\centerline {\bf Math 523a - Homework 5} 

\bigskip

\begin{enumerate}

\item Folland, problem 46, p 68.

\item Folland, problem 59, p 77. 

\item Let $(X,\M,\mu)$ be a $\sigma$-finite measure space. 
Let $f:X \ra \reals$ 
be a non-negative, measurable function. 
Define a subset of $X \times \reals$ by 
\bea
E=\{ (x,t):f(x) \ge t \ge 0 \}
\eea
We use the product $\sigma$-algebra on $X \times \reals$, i.e., 
$\M \otimes \B$ where $\B$ is the Borel $\sigma$-field in $\reals$. 
In the last homework you proved that $E$ is in $\M \otimes \B$.
Let $m$ denote Lebesgue measure on $\reals$. Then $\mu \times m$ is 
a measure on $(X \times \reals,\M \otimes \B)$.
Prove that
\bea
\mu \times m (E) = \int f \, d\mu = \int_{[0,\infty)} F(t) dm 
\eea
where $F(t) = \mu(\{x:f(x) \ge t\})$. 
Hint: $\mu \times m (E) = \int \chi_E \, d \mu \times m$. 

\item 
Let $\mu$ be a {\bf finite} Borel measure on $\reals$ such that 
$\int_\reals \sqrt{|x|} \, d \mu < \infty$. 
Prove that there is a set $N \subset \reals$ of  
Lebesgue measure zero such that for $s \notin N$, 
\bea
\int_\reals |s-t|^{-1/2} \, d\mu(t) < \infty 
\eea
Hint: Let $L>0$. 
Prove that the above is finite when $|s| \le L$, except for 
a set of Lebesgue measure zero. For this consider
\bea
\int_{[-L,L]} \left[ \int_\reals |s-t|^{-1/2} \, d\mu(t) \right] dm(s)
\eea

\item 
Let $m$ denote Lebesgue measure on $\reals^n$. 
For this problem theorem 2.49 on p. 78 may be useful. 
(We didn't prove this in class, but you can use it.)
\begin{enumerate}
\item 
Let 
\bea
f(x_1,x_2,\cdots,x_n) = \left[\sum_{i=1}^n x_i\right]^{-p} 
\eea
Determine (with justification) for what values of $p$ we have 
$f \in L^1([0,1]^n,m)$. 
\item 
Let 
\bea
g(x_1,x_2,\cdots,x_n) = \left[1 + \sum_{i=1}^n x_i\right]^{-p} 
\eea
Determine (with justification) for what values of $p$ we have 
$f \in L^1([0,\infty)^n,m)$. 
\end{enumerate}

\end{enumerate}

\end{document}
\end


