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\textbf{MATH 501, Analysis, Homework 2}
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\underline{\textbf{1. (3 pts)}}\footnote{This is Folland's Exercise 1.18.b. You may want to take a look at the other parts of the question to get a hint.} Let $\mathcal{A} \subseteq \mathcal{P}(X)$ be an algebra on a set $X$ and $\rho: \mathcal{A} \rightarrow [0,+\infty]$ be a premeasure on $X$. Recall that the map $\mu^*: \mathcal{P}(X) \rightarrow [0,\infty]$ given by
\[ \mu^*(S)=\inf\left\{\sum_{i=1}^{\infty} \rho(A_i): A_i \in \mathcal{A},\ S \subseteq \bigcup_{i=1}^{\infty} A_i\right\}\]
is an outer measure on $X$. Let $E \subseteq X$ be such that $\mu^*(E) < \infty$. Show that $E$ is $\mu^*$-measurable if and only if there exists a set $A \in \mathcal{M}(\mathcal{A})$ such that $E \subseteq A$ and $\mu^*(A-E)=0$.\\

\underline{\textbf{2. (2 pts)}} Let $F(x)=\lfloor x \rfloor$ be the floor function. Consider the Lebesgue-Stieltjes measure $\mu_F: \mathcal{B}(\mathbb{R}) \rightarrow [0,\infty]$ associated to $F$. Show that there exists a co-countable open set $S \subseteq \mathbb{R}$ which is $\mu_F$-null. \\

\underline{\textbf{3. (2 pts)}} Prove that there exists an uncountable closed set $D \subseteq \mathbb{R}$ with the property that $D$ contains no open intervals and $\mathbf{m}(D)=\infty$, where $\mathbf{m}$ is the usual Lebesgue measure on $\mathbb{R}$.\\

\underline{\textbf{4. (3 pts)}}\footnote{In this question, we shall prove \textbf{Steinhaus's theorem} which states that $A-A$ contains an open interval centered at the origin for any Lebesgue measurable set $A$ with positive measure.} Let $A \subseteq \mathbb{R}$ be a Lebesgue measurable set such that $0<\mathbf{m}(A)<\infty$ where $\mathbf{m}$ is the usual Lebesgue measure on $\mathbb{R}$.
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\item[a.] Prove that there exist a closed set $F \subseteq A$ and an open set $A \subseteq G$ such that we have $3\mathbf{m}(G) < 4 \mathbf{m}(F)$.
\item[b.] Let $G$ be as in part a. Prove that there exists an open interval $I \subseteq G$ such that $3\mathbf{m}(I) < 4 \mathbf{m}(F \cap I)$.
\item[c.] Let $I$ be as in part b. Let $\delta=\mathbf{m}(I)/2$. Show that
$$B(0,\delta) \subseteq A-A=\{a-b: a,b \in A\}$$
(Keep in mind that $(F\cap I) \cup (x+(F\cap I)) \subseteq I \cup (x+I)$. Now try to argue that $F \cap I$ and $x+(F \cap I)$ cannot be disjoint if $|x|<\delta$. Then finish the proof!)\\
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\textbf{Solutions}
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