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\begin{document}
\title{Math 101 Homework}
\author{Mary Radcliffe}
\date{due 8 Feb 2015}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Let $X=\{1, 2, 3, \dots, 2n\}$. Let $\mathcal{F}$ be the collection of subsets of $X$ that have even size. For example, if $X=\{1, 2, 3, 4\}$, then 
\[\mathcal{F} = \left\{\emptyset, \{1, 2\}, \{1, 3\}, \{1, 4\}, \{2, 3\}, \{2, 4\}, \{3, 4\}, \{1, 2, 3, 4\}\right\}.\]

Is $\mathcal{F}$ a $\sigma$-field? If yes, prove it. If not, which properties does it satisfy?

\item Let $X$ be a set, and define
\[\mathcal{F} = \{S\subset X\ | \ \hbox{either }S\hbox{ is at most countable or }S^c\hbox{ is at most countable}\}.\]

Prove that $\mathcal{F}$ is a $\sigma$-field. You may use the fact that the union of countably many countable sets is countable.

\item In the definition of a $\sigma$-field, we gave some properties in terms of set operations. Here are some more set operations we can do in a $\sigma$-field.

Suppose $\mathcal{F}$ is a $\sigma$-field. Prove each of the following:
\begin{enumerate}
\item $\emptyset\in \mathcal{F}$
\item If $A_1, A_2, A_4, \dots\in \mathcal{F}$, then $\displaystyle\bigcap_{i=1}^\infty A_i\in\mathcal{F}$. (Hint: Use De Morgan's Law)
\item If $A, B\in \mathcal{F}$, then $A\ba B\in \mathcal{F}$.
\end{enumerate}

\item Suppose that $\mu$ is a measure on a $\sigma$-field $\mathcal{F}$. 

\begin{enumerate}

\item Prove that if $A, B$ are disjoint sets in $\mathcal{F}$, then $\mu(A\cup B) = \mu(A)+\mu(B)$.
\item Prove that if $A_1, A_2, \dots\in \mathcal{F}$ are pairwise disjoint, then 
\[\mu(\cup_{i=1}^\infty A_i) = \sum_{i=1}^\infty \mu(A_i).\]
\end{enumerate}

\end{enumerate}









\end{document}