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\textbf{MATH 501, Analysis, Homework 1}
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\underline{\textbf{1. (3 pts)}} Let $X$ be a non-empty set and $\mathcal{M} \subseteq \mathcal{P}(X)$ be a $\sigma$-algebra on $X$ generated by $\mathcal{E} \subseteq \mathcal{M}$. Show that 
\[ \mathcal{M}=\bigcup_{F \in \mathcal{P}_{c}(\mathcal{E})} \mathcal{M}(F) \]
where $\mathcal{M}(F)$ denotes the $\sigma$-algebra generated by $F$ and $\mathcal{P}_{c}(\mathcal{E})$ denotes the set of countable subsets of $\mathcal{E}$, that is, $\mathcal{P}_{c}(\mathcal{E})=\{F \subseteq \mathcal{E}: F \text{ is countable}\}$.\\

\underline{\textbf{2. (2 pts)}} Show that the set
\[ \{(x,y) \in \mathbb{R}^2: x^7+xy+y^7 \notin \mathbb{Q}\}\]
is a Borel subset of $\mathbb{R}^2$. (\textbf{Hint.} Recall that the inverse images of open sets are open under continuous mappings.)\\

\underline{\textbf{3. (2 pts)}} Let $(X, \mathcal{M})$ be a measurable space and $\mu: \mathcal{M} \rightarrow [0,\infty]$ be a \textit{finitely additive} measure on $(X, \mathcal{M})$. Suppose that if $A_1,A_2,\dots \in \mathcal{M}$ with $A_1 \subseteq A_2 \subseteq \dots$, then $\mu(\bigcup_{n=1}^{\infty} A_n)=\lim_{n \rightarrow \infty} \mu(A_n)$. Prove that $\mu$ is indeed a ($\sigma$-additive) measure.\\

\underline{\textbf{4. (3 pts)}} Let $X$ be an uncountable set and $\mathcal{M}=\{A \subseteq X: A \text{ or } A^c \text{ is countable}\}$. Recall that the triple $(X,\mathcal{M},\eta)$ is a measure space where $\eta: \mathcal{M} \rightarrow [0,\infty]$ is given by
\[ \eta(A)=\begin{cases} 1 & \text{ if } A^c \text{ is countable}\\ 0 & \text{ if } A \text{ is countable}\end{cases}\]
Consider the relation $\sim$ on $\mathcal{M}$ given by
\[ A \sim B \Leftrightarrow \eta(A \triangle B)=0\]
Show that $\sim$ is an equivalence relation and find the cardinality of $\mathcal{M}/\sim$.

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\textbf{Solutions}
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