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\begin{document}
\begin{center}
{\bf \large Math 702 Problem Set 8} \vs \\
{\bf due Friday 4/11/2014} \vs \vs
\end{center}

\noindent
In the following problems, $(X,\mu)$ is always a measure space,
$$
\| f \|_p = \left( \int_X |f|^p \, d\mu \right)^{1/p} \qquad (0<p<\infty),
$$ 
and $L^p=L^p(\mu)$ is the space of all measurable $f: X \to \CC$ for which $\| f \|_p<\infty$. \vs \vs

\noindent
{\bf Problem 1.} Let $f: X \to \CC$ be measurable. Prove the following statements: \vs
\begin{enumerate}
\item[(i)]
If $0<r<p<s \leq \infty$ and $f \in L^r \cap L^s$, then $f \in L^p$. In other words, the set $\{ p \in (0,\infty] : f \in L^p \}$ is connected. \vs

\item[(ii)]
If $0<r<s \leq \infty$ and $f \in L^r \cap L^s$, and if $f$ is not a.e. equal to $0$, the function $\varphi(p) = \log \| f \|_p^p$ is convex in the interval $(r,s)$. \vs
\end{enumerate}
(Hint: For (i), consider the sets where $|f|<1$ and $|f| \geq 1$. For (ii), take $x,y \in (r,s)$ and $0<t<1$ and use H\"{o}lder's inequality for $p=1/t, q=1/(1-t)$ to show $\varphi((1-t)x+ty) \leq (1-t)\varphi(x)+t \varphi(y)$.) \vs

\noindent
{\bf Problem 2.} Suppose $f: X \to \CC$ is measurable and $f \in L^r$ for some $0<r<\infty$. Prove the following statements about the behavior of $\| f \|_p$ for large and small $p$: \vs

\begin{enumerate}
\item[(i)]
$\lim_{p \to \infty} \| f \|_p = \| f \|_{\infty}$ whether or not $\| f \|_{\infty}$ is finite. \vs

\item[(ii)]
If $\mu(X)=1$, then 
$$
\lim_{p \to 0} \| f \|_p = \exp \left( \int_X \log |f| \, d\mu \right)
$$
if we define $\exp(-\infty)=0$. \vs
\end{enumerate}
(Hint: For (ii), use Jensen's inequality for one direction, and $\log x \leq x-1$ together with $\lim_{p \to 0}(|f|^p-1)/p=\log |f|$ for the other direction.) \vs 

\noindent
{\bf Problem 3.} Suppose $\mu(X)<\infty$, $f \in L^{\infty}(\mu)$, and $\| f \|_{\infty}>0$. Define
$$
\alpha_n = \int_X |f|^n \, d\mu, \qquad \ n=1,2,3,\ldots
$$
Show that 
$$
\lim_{n \to \infty} \frac{\alpha_{n+1}}{\alpha_n} = \| f \|_{\infty}. \vs
$$

\noindent
{\bf Problem 4.} For each $0<p\leq \infty$, find a function in $L^p(\RR)$ that does not belong to $L^r(\RR)$ for any $r \neq p$. \vs 

\noindent
{\bf Problem 5.} The {\bit essential range} of $f \in L^{\infty}(\mu)$ is the set $R_f$ of points $z \in \CC$ for which $\mu( \{ x \in X: |f(x)-z|< \ve \})$ is positive for every $\ve>0$. Show that $R_f$ is compact. What relation can you establish between $R_f$ and $\| f \|_{\infty}$? \vs 

\noindent
{\bf Problem 6.} A sequence $f_n : X \to \CC$ of measurable functions is said to {\bit converge in measure} to a measurable function $f: X \to \CC$ if for every $\ve>0$,
$$
\mu \big( \{ x \in X: |f_n(x)-f(x)|>\ve \} \big) \to 0 \quad \text{as} \quad n \to \infty. 
$$
Prove the following statements: \vs
\begin{enumerate}
\item[(i)]
If $\mu(X)<\infty$ and $f_n \to f$ a.e., then $f_n \to f$ in measure. \vs
\item[(ii)]
If $f_n \to f$ in $L^p(\mu)$ for some $1 \leq p \leq \infty$, then $f_n \to f$ in measure. \vs 
\item[(iii)]
If $f_n \to f$ in measure, some subsequence of $f_n$ converges to $f$ a.e. \vs 
\end{enumerate}  
Show that the converses of (i) and (ii) do not hold, so convergence in measure is weaker than both a.e. pointwise convergence and $L^p$ convergence. 
(Hint: For (i), use Egoroff's theorem. For (iii), the Borel-Cantelli lemma is useful.) 
\end{document}

