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\begin{center}
{\huge  \sc  Math 201a: Homework 5} \\
{\large  Due Sunday, November 10th at 11:59pm}
\end{center}
Questions followed by * are to be turned in. Questions without * are extra practice. At least one extra practice questions will appear on each exam. \emph{All answers should be justified with either a proof or a counterexample.
}
 \section*{Question 1*} %HW6Q1


Consider a measure space $(X,\mathcal{M}, \mu)$. Suppose $f:X \to \overline{\mathbb{R}}$ is measurable.
\begin{enumerate}[(a)]
\item If $f \geq 0$ and $\int f d \mu < +\infty$, prove that $\{ x : f(x) = +\infty \}$ is a null set.
\item Under the same hypotheses of part (a), prove that $ \{ x : f(x) >0 \}$ is $\sigma$-finite.
\item If $f \in L^1(\mu)$, prove that there exists $g: X \to \mathbb{R}$ so that $g=f$ $\mu$-a.e.. 
\end{enumerate}

 
\section*{Question 2*} %formerly HW6Q3
Let $\mu$ be a locally finite measure on $(\R, \mathcal{B}_\R)$. (See HW4, Q1 for the definition and some key facts.) Define
\[ B(x,r) = \{ y: |y-x| < r \}, \quad \bar{B}(x,r)=  \{ y: |y-x| \leq r \} . \]
\begin{enumerate}[(a)]
\item Prove that, for all $r>0$, $x \mapsto \mu(B(x,r))$ is lower semicontinuous. \\
\emph{Hint: First, prove that $\mu(B(x,r)) = \lim_{n \to +\infty} \mu(B(x,r-1/n))$.}
\item  Prove that, for all $r >0$, $x \mapsto \mu(\bar{B}(x,r))$ is upper semicontinuous. \\
\emph{Hint: How can you adapt the hint from part (a)?}
\item For any $k \geq 0$, prove that 
\[ f(x) = \limsup_{r \to 0} \frac{ \mu(B(x,r))}{r^k} \]
is Borel measurable. \\
\emph{Hint: Write $f(x) = \lim_{r \to 0} G_r(x)$, where 
\[G_r(x) = \sup \left\{  \frac{ \mu(B(x,\rho))}{\rho^k} : 0 < \rho < r \right\} . \]}
\end{enumerate}

\section*{Question 3}
\begin{enumerate}[(a)]
\item Suppose $\phi: \R \to \R$ is Lebesgue measurable, $\psi: \R \to \R$,  and $\phi=\psi$ Lebesgue almost everywhere. Then $\psi$ is Lebesgue measurable.
\item Now, suppose $\phi: \R \to \R$ is Borel measurable, $\psi: \R \to \R$,  and $\phi=\psi$ Borel almost everywhere. Is $\psi$ is Borel measurable? Justify your answer with a proof or counterexample.
\end{enumerate}




 \section*{Question 4*} %formerly HW6Q5
 \begin{enumerate}[(a)]
 \item Given a Lebesgue measurable function $f: \R \to \R$ and a Lebesgue measurable set $E$ prove that
 \begin{align*}
\int_E f(x) d \lambda(x) = \int_{E+c} f(x-c) d \lambda(x) \quad \text{ for all } c \in \R,
\end{align*}
whenever the integral $\int_E f(x) d \lambda(x) $ exists.
\item Suppose that $f: \R \to \R$ is a nondecreasing function that is differentiable Lebesgue almost everywhere. (In fact,  you can  prove in a later measure theory course that all nondecreasing functions are differentiable Lebesgue-almost everywhere!) Prove that 
\begin{align*}
\int_{[0,1]} \limsup_{n \to +\infty} \frac{f(x +1/n) - f(x)}{1/n} \  d \lambda(x) \leq f(1) - f(0)
\end{align*}
\emph{Hint: It can be convenient to assume, without loss of generality, that $f(x) = f(1)$ for all $x \geq 1$. Use part (a).}
\end{enumerate}


\section*{Question 5}%formerly HW6Q6 
Consider a measure space $(X, \mathcal{M}, \mu)$ and a sequence of nonnegative, measurable functions $f_n$ satifying 
\[ f_n(x) \searrow f(x) \text{ for all }x \in X \quad \text{ and }  \quad\int f_1 < +\infty. \] Prove that 
\[ \lim_{n \to +\infty} \int f_n = \int \lim_{n \to +\infty} f_n . \]


\section*{Question 6*}%formerly HW6Q7
Consider a measure space $(X, \mathcal{M}, \mu)$ and a nonnegative, measurable function $f \in L^1(\mu)$. Prove that, for all $\epsilon >0$, there exists $A \in \mathcal{M}$ so that $\mu(A) < +\infty$ and $\int_A f > \int f - \epsilon$. \\ 

\emph{Hint: use the definition of the integral in terms of simple functions, and choose a simple function with integral close to $\int f d \mu$.}

\section*{Question 7*}%formerly HW6Q8
Prove the following \emph{generalized dominated convergence theorem}:

\begin{thm}
Consider a measure space $(X, \mathcal{M}, \mu)$. Suppose $f_n, g_n, f, g \in L^1(\mu)$ are functions for which the following hold:
\begin{enumerate}[(i)]
\item $f_n \to f $ $ \mu$-a.e. ,
\item $g_n \to g$ $\mu$-a.e. ,
\item $|f_n| \leq g_n$ $\mu$-a.e. ,  
\item $\int g_n d \mu \to \int g d \mu$.
\end{enumerate}
Then $\int f_n d \mu \to \int f d \mu$.
\end{thm}

\section*{Question 8*} %Formerly HW6 Q7
 Suppose $(X, \mathcal{M}, \mu)$ is a measure space and $f : X \to [0,+\infty]$ satisfies $f \in L^1(\mu)$. Prove that, for all $\epsilon >0$, there exists $\delta >0$ so that if $E \in \mathcal{M}$ satisfies $\mu(E) < \delta$, then $\int_E f d \mu < \epsilon$. \\ \ \\
 \emph{Hint: Proceed by contradiction, so that you can find a sequence $E_n \in \mathcal{M}$ with arbitrarily small measure so that $\int_{E_n} f d \mu \geq \epsilon$. Consider the set $A = \cap_{n =1}^{+\infty} \cup_{k > n} E_k$.}


\section*{Question 9}
Consider a measure space $(X, \mathcal{M}, \mu)$.
\begin{enumerate}[(a)]
\item Fix a nonnegative, measurable function $f$. Prove that $\nu(A) := \int_A f d \mu$ is a measure on $\mathcal{M}$.
\item Suppose $(X, \mathcal{M}, \mu) = (\mathbb{R}, \mathcal{B}_\mathbb{R}, \lambda)$, where $\lambda$ is Lebesgue measure. Fix $f \in L^1(\lambda)$. Consider the cumulative distribution function
\[  F(x):= \int_{(-\infty,x]} f d \lambda  .\]
Prove that $F$ is continuous.
\item Given an example of a Borel measure $\mu$ on the real line  for which the cumulative distribution function
\[ F(x):=  \int_{(-\infty, x]}  d \mu . \]
 is \emph{not} continuous.
\end{enumerate}

\section*{Question 10}
In this question, you will show why the $\liminf$ in Fatou's lemma can't be naively replaced with a $\limsup$. Fix Give an example of a measure space $(X,\mathcal{M},  \mu)$ and sequence of nonnegative measurable functions $\{f_n\}_{n \in \mathbb{N}}$ so that $f_n$ converges pointwise to some $f$ but
\[ \limsup_{n \to +\infty} \int f_n d \mu < \int \limsup_{n \to +\infty} f_n d \mu . \]


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