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\begin{center}
{\huge  \sc  Math 201a: Homework 4} \\
{\large  Due Sunday, November 3rd 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}
  Suppose $\mu$ is a \emph{locally finite Borel measure} on $\R$, that is  $\mu(B)<+\infty$ for all balls $B \subseteq \mathbb{R}$.
\begin{enumerate}[(i)]
\item  Prove that, for all $S \in \mathcal{B}_\R$ and $\epsilon >0$, there exists an open set $U$ satisfying $S\subseteq U$ so that $\mu(U \setminus S) \leq \epsilon$.
\item  Prove that, for all $S \in \mathcal{B}_\R$ and  $\epsilon >0$ there exists a closed set $K \subseteq S$ so that $\mu(S \setminus K) \leq \epsilon$.
\item As a consequence of these results, prove that any locally finite measure is inner and outer regular. (This can also be seen as a consequence of Theorem 1.16 in Folland, which shows all locally finite Borel measures on $\R$ are Lebesgue Stieltjes measures.)
\end{enumerate}
 
 
 
\section*{Question 2}  
Let $(X, \mathcal{M})$ be a measurable space.
 \begin{enumerate}[(i)]
 \item  Consider $f: X \to \mathbb{R}$.  Prove that the following are equivalent:
 \[ f \text{ is }(\mathcal{M}, \mathcal{B}_{\overline{\mathbb{R}}}) \text{ measurable } \iff
f \text{ is }(\mathcal{M}, \mathcal{B}_{{\mathbb{R}}}) \text{ measurable.} \] % {\color{red}double check -- new question}
\item Consider $f: X \to \overline{\R}$.  Prove that $f$ is measurable if and only if $f^{-1}(\{-\infty\}) \in \mathcal{M}$, $f^{-1}(\{+\infty\}) \in \mathcal{M}$, and $f^{-1}(B) \in \mathcal{M}$ for all $B \in \mathcal{B}_\R$. % {\color{cyan}formerly Midterm 1, Q1b}
 \end{enumerate}
 

\section*{Question 3*}
The Cantor set may be defined as follows: define $C_0 = [0,1]$, $C_1 = C_0 \setminus (\frac13, \frac23) = [0,\frac13]\cup[\frac23,1]$, .... where $C_n$ is formed from $C_{n-1}$ by removing the open middle third of each of the finitely many closed intervals making up $C_{n-1}$. This process is illustrated nicely in the following picture from Wikipedia: \\

\includegraphics[width = 5in]{Cantorset.png} \\

 The Cantor set is defined by $C = \cap_{n=1}^{+\infty} C_n$. \\ 
 
 The Cantor set is a canonical example in  because it is ``large'' in cardinality but ``small'' both in measure and topologically. In particular, one can show that it is ``large'' because it has cardinality of the continuum (see Prop 1.22 in Folland). In this question, you will prove the sense in which it is ``small.''
 \begin{enumerate}[(a)]
 \item  Prove that $C_n$ is the union of $2^n$ disjoint closed intervals, each of length $(1/3)^n$. (This follows quickly from the above construction.)
 \item Show that, if $x,y \in C$ and $x<y$, then there exists $z \in (x,y)$, $z \not \in C$. In other words, the Cantor set does not contain any intervals. (Since the only connected subsets of $C$ are single points, we say $C$ is \emph{totally disconnected}.)
 \item Show that the Cantor set is closed.
 \item Show that the Cantor set is nowhere dense, that is ${\rm int}({\rm cl}(C)) = \emptyset$. (This implies it is ``small'' topologically.)
  \item Show that $\lambda(C) = 0$. (This implies it is ``small'' in measure.)
 \end{enumerate}




\section*{Question 4}
  Consider topological spaces $X$ and $Y$. Suppose  $X$ is compact and $f: X \to Y$ is continuous and invertible. Prove that $f^{-1}$ is continuous.


\section*{Question 5*}% formerly Q6
The \emph{Cantor function} $f:[0,1] \to [0,1]$ may be defined as follows. For each $x \in (\frac13,\frac23)$, let $f(x) = \frac12$. Thus $f$ is defined at each point removed from $C_0$ in the construction of $C_1$. Next, we define $f$ at each point removed from $C_1$ in the construction of $C_2$ by letting $f(x) = \frac14$ if $x \in (\frac19, \frac29)$ and $f(x) = \frac34$ if $x \in (\frac79, \frac89)$. Continuing in this way, we define $f(x)$ to be $\frac{1}{2^n}$, $\frac{3}{2^n}$, $\frac{5}{2^n}$, ...  $\frac{2k-1}{2^n}$, ..., for $k =1, \dots, 2^{n-1}$, on the various intervals removed from $C_{n-1}$ in the construction of $C_n$. This prescribes the values of  $f$ on the open set $[0,1] \setminus C$. By definition, this function  is non-decreasing and   strictly   bounded between 0 and 1 on this set. Finally, we define $f$ on the entire interval $[0,1]$ by letting $f(0)=0$ and, for $x \in C \setminus \{0\}$,
\begin{align*}
f(x) = \sup_{t<x} \{f(t) : t \in [0,1] \setminus C\}.
\end{align*}
\begin{enumerate}[(a)]
\item Prove that $f$ is nondecreasing, continuous, and that $f(1) = 1$.
\item Prove that $g(x) = f(x) +x$ is a bijection from $[0,1]$ to $[0,2]$.
\item Prove that $g^{-1}(x)$ is continuous.
\item Prove that $\lambda(g(C)) = 1$.
\item On HW3 Q9, you   showed that Lebesgue measure on $\mathcal{M}_{\lambda^*}$ is \emph{complete}, that is, every subset of a Lebesgue null set must be Lebesgue measurable. Use this fact to prove that $g(C)$ contains  a set $A \not \in \mathcal{M}_{\lambda^*}$ so that $g^{-1}(A) \in \mathcal{M}_{\lambda^*} \setminus \mathcal{B}_\R$. \emph{Hint: use HW3 Q5 and Q9.}
\item Use part (e) to prove that Lebesgue measure on the Borel sets is \emph{not} complete.
\end{enumerate}
The main point of this question is that parts (e-f) demonstrate $\mathcal{B}_\R \subsetneq \mathcal{M}_{\lambda^*}$.
  
  \section*{Question 6* }%{\color{red}formerly HW5Q4}}
 Give an example of an \emph{uncountable} collection of Borel functions such that the pointwise supremum is \emph{not} a Borel function.  \\  \ \\
\emph{Hint: consider the indicator function on a nonmeasurable set $A$, defined by}
\begin{align*}
1_A(x) = \begin{cases} 1 &\text{ if } x \in A , \\ 0 &\text{ if } x \not \in A . \end{cases}
\end{align*}



\section*{Question 7}
%{\color{red}This was HW6Q2, but it makes mroe sense earlier}
 \begin{lem}
If $\mu$ is a Lebesgue-Stieltjes measure on $\mathbb{R}$ and $E \in \mathcal{M}_{\mu^*}$ with $\mu(E) < +\infty$, then for all $\epsilon >0$, there exists a finite collection of disjoint open intervals $\{ I_i\}_{i=1}^n$ so that 
\[ \mu \left( E \Delta \left( \cup_{i=1}^n I_i \right) \right) < \epsilon . \]
 \end{lem}
 Prove the lemma.


\section*{Question 8*}
Consider a measurable space $(X \mathcal{M})$. Given $f, g : X \to  \bar{\mathbb{R}}$ measurable, prove the following:
\begin{enumerate}[(a)]
\item $fg$ is measurable (where $0 \cdot \pm \infty = 0$)
\item Fix $a \in \bar{\mathbb{R}}$ and define $h(x) = a$ if $f(x) = - g(x) = \pm \infty$ and $h(x) = f(x)+g(x)$ otherwise. Then $h$ is measurable.
\end{enumerate}

 
\section*{Question 9*}%formerly HW5Q3
Let $f: \R \to \R$ be a continuous function. 
\begin{enumerate}[(a)]
\item  Prove that 
\[  g(x) = \limsup_{h \to 0} \frac{ f(x+h) - f(x)}{h} \]
is Borel measurable. (Hint: Recall HW1, Q8(a) and Midterm 1, Q3(a).) \\
\item Let $D$ be the set of points at which $f$ is differentiable. Prove that $D \in \mathcal{B}_\R$. 
\end{enumerate}


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