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\begin{center}
{\huge  \sc  Math 201a: Homework 2} \\
{\large  Due Sunday, October 13th 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.

\section*{Question 1}
Prove the following:
\begin{enumerate}[(a)]
\item  Any algebra that is closed under countable disjoint unions is a $\sigma$-algebra.
\item Any algebra that is closed under countable increasing unions is a $\sigma$-algebra. (We say that an algebra $\mathcal{A}$ is closed under countable increasing unions if, for all $\{ E_i\}_{i=1}^\infty \subseteq \mathcal{A}$ with $E_i \subseteq E_{i+1}$ for all $i$, $\cup_{i=1}^\infty E_i \in \mathcal{A}$.)
\end{enumerate}


\section*{Question 2}
\begin{enumerate}[(a)]
\item Prove the following lemma from lecture 2:
\begin{lem}
Suppose $\mathcal{A}$ is an algebra of subsets of a set $X$.
\begin{enumerate}[(i)]
\item If $E_1, \dots, E_n \in \mathcal{A}$, then $\cap_{i=1}^n E_i \in \mathcal{A}$.
\item $\emptyset \in \mathcal{A}$ and $X \in \mathcal{A}$.
\end{enumerate}
\end{lem}
\item Prove the following claim from lecture 2:

\textbf{Claim:}\emph{ Given a nonempty collection $\mathcal{C}$ of $\sigma$-algebras on $X$, 
\[ \cap \mathcal{C} := \{ E: E \in \mathcal{A} , \ \forall \mathcal{A} \in \mathcal{C} \} \]
 is a $\sigma$-algebra.}
\end{enumerate}

\section*{Question 3}
Fix an open set $U \subseteq \R$. Prove that there exist $\{ a_n \}_{n=1}^\infty, \{b_n \}_{n=1}^\infty \subseteq \R$ so that $U = \cup_{n =1}^\infty (a_n, b_n)$.

\section*{Question 4*}
Prove the following proposition from lecture 2:
\begin{prop}
The Borel $\sigma$-algebra of $\R$ is generated by each of the following:
\begin{enumerate}[(i)]
\item the half-open intervals: $\mathcal{E}_3  = \{ (a,b] : a<b \}$,
\item the open rays: $\mathcal{E}_5 = \{ (a,+\infty): a \in \mathbb{R} \}$,
\item the closed rays: $\mathcal{E}_7 = \{ [a, +\infty) : a \in \R \}$.
\end{enumerate}
\end{prop}
 
\section*{Question 5*}
Let $X$ be an uncountable set, and let $\mathcal{A}$ be the collection of subsets $E \subseteq X$ such that either $E$ or $E^c$ is at most countably infinite. Prove that $\mathcal{A}$ is a $\sigma$-algebra.

\section*{Question 6*}
 Given a collection of sets $\{ E_i\}_{i=1}^\infty$, we may define
\begin{align*}
\limsup_{i \to +\infty} E_i := \cap_{k=1}^\infty \cup_{i=k}^\infty E_i \ , \qquad \liminf_{i \to +\infty} E_i := \cup_{k=1}^\infty \cap_{i=k}^\infty E_i \  .
\end{align*}
We may also consider the sets
\begin{align*}
A_1:= \{x : x\in E_i \text{ for all but finitely many $i$} \} \ , \qquad A_2:= \{ x : x \in E_i \text{ for infinitely many $i$} \} \ .
\end{align*}
Determine for which values of $n, m \in \{1,2\}$, $\limsup_{i \to +\infty} E_i = A_n$ and $\liminf_{i \to +\infty} E_i = A_m$. Justify your answer with a proof.

\section*{Question 7}
Each $x \in [0,1]$ has a base-10 decimal expansion $x = \sum_{j=1}^{+\infty} a_j 10^{-j}$, where $a_j = 0, 1, \dots, 9$. This expansion is unique unless $x$ is of the form $p10^{-k}$, for $p , k \in \mathbb{N}$, where $p$ is not divisible by $10$. In this case $x$ has two expansions: one with $a_j = 0$ for $j>k$ and one with $a_j = 9$ for $j >k$. For the purposes of this problem, suppose that we always choose the \emph{standard decimal representation of $x$}, which is the one for which $a_j = 0 $ for $j >k$.

\begin{enumerate}[(a)]
\item Fix $n \in \mathbb{N}$. Let $A_n$ be the set of numbers $x$ in the interval $[0,1]$ with a 7 in the $n$th decimal place. Prove that $A_n$ belongs to the Borel $\sigma$-algebra.
\item  Let $E$ be the set of numbers $x$ in the interval $[0,1]$ so that the decimal expansion for $x$  contains a 7. Prove that $E$ belongs to the Borel $\sigma$-algebra.
\item Let $F$ be the set of numbers $x$ in the interval $[0,1]$ so that the decimal expansion for $x$ contains infinitely many 7's. Prove that $F$ belongs to the Borel $\sigma$-algebra. 
\end{enumerate}



 

\section*{Question 8*} %formerly HW2 Q9
Consider a metric space $(X,d)$ and $f : X \to \overline{\R}$. Recall the notion of lower semicontinuous envelope $f_*$ from Homework 1. Let $f^*$ denote the upper semicontinuous envelope of $f$, which is defined symmetrically. You may use all the analogous properties of the upper semicontinuous envelope $f^*$ that you proved for the lower semicontinuous envelope $f_*$ on Homework 1.
\begin{enumerate}[(a)]
\item Prove that $f$ is continuous at $x \in X$ if and only if $f_*(x) = f^*(x) = f(x)$.
\item Let $E$ be the set of points at which $f$ is discontinuous. Prove that $E$ is a countable union of closed sets, hence a Borel set. (\emph{Hint}: What can you say about the set $\{y: f_*(y) \geq f^*(y) \}$?)
\end{enumerate}

 
\section*{Question 9*} %formerly HW3 Q6
Suppose $(X, \mathcal{M}, \mu)$ is a measure space with $\mu(X) <+\infty$. Suppose $A_1, A_2, \dots$ are sets in $\mathcal{M}$ with $\mu(A_i) \geq c >0$ for all $i$. Let $Z$ be the set of elements $x \in X$ that belong to infinitely many of the $A_i$'s. Prove that $\mu(Z)  \geq c$.
 

\section*{Question 10}
Given a measure space $(X, \mathcal{M}, \mu)$ and $E \in \mathcal{M}$ nonempty, define \emph{the restriction of $\mu$ to $E$} by $\mu_E(A):=\mu(A \cap E)$ for all $A \in \mathcal{M}$. Prove that $\mu_E$ is a measure on $\mathcal{M}$.


\section*{Question 11*} %formerly HW3Q1
Suppose that $X$ is an uncountable set. Define outer measures $\mu^*$ and $\nu^*$ on $X$ by 
\begin{align*}
\mu^*(E) = \begin{cases} 0 &\text{ if $E$ is countable} , \\ 1 &\text{ if $E$ is uncountable,} \end{cases} \qquad \nu^*(E) = \begin{cases} 0 &\text{ if $E$ is countable,} \\ +\infty &\text{ if $E$ is uncountable.} \end{cases} 
\end{align*}
\begin{enumerate}[(a)]
\item Find all sets that are $\mu^*$ measurable. Justify your answer.
\item Find all sets that are $\nu^*$ measurable. Justify your answer.
\end{enumerate}



\section*{Question 12*}
By Caratheodory's Theorem, we know that any outer measure $\mu^*$ can lead to a measure by restricting $\mu^*$ to the collection of $\mu^*$-measurable sets. However, in this problem, we will show that the collection of $\mu^*$-measurable sets is, in general, not the largest $\sigma$-algebra on which the restriction of $\mu^*$ becomes a measure. \\ 
Consider the set $X = \{1,2,3\}$. Define an outer measure as follows:
\begin{align*}
\mu^*(A) = \begin{cases} 0 & \text{ if } |A| = 0 , \\ 1 & \text{ if } |A| =1 ,2 \\  2 &\text{ if } |A| = 3. \end{cases}
\end{align*}
\begin{enumerate}[(a)]
\item Prove that $\mu^*$ is an outer measure on $X$.
\item Prove that the collection of $\mu^*$ measurable sets is $\{ \emptyset , X\}$. 
\item Prove that $\mathcal{A} := \{ \emptyset , \{1\}, \{2,3\}, X \}$ is a $\sigma$-algebra.
\item Prove that $\mu^*|_{\mathcal{A}}$ is a measure.
\end{enumerate} 
This shows that the Caratheodory $\sigma$-algebra $\mathcal{M}_{\mu^*}$ is not, in general, the largest $\sigma$-algebra on which $\mu^*$ can be restricted to be a measure.

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