\documentclass[12pt]{amsart}
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\newtheorem*{definition}{Definition}
\newtheorem*{example}{Example}
\newtheorem*{exercise}{Exercise}
\newtheorem*{conjecture}{Conjecture}
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\newtheorem*{lemma}{Lemma}
\newtheorem*{remark}{Remark}
\begin{document} 
\title{Assignment 2}
\section{Assignment 2}
The coin toss experiment has two outcomes, H and T. In the following problems, the experiment consists of repeating the coin toss until the first heads is obtained.  The assignment consists of constructing a probability triple $(S,\Sigma,\mu)$ that represents the experiment.  
\subsection{Problem 1}
\par\vspace{0.3 cm}
Let $S$ be the set of all possible outcomes of the experiment.  Write a definition for $S$.
\subsection{Problem 2}
\par\vspace{0.3 cm}
Define a $\sigma$-algebra $\Sigma$ on $S$, that is, a collection $\Sigma$ of subsets that is closed (stable) under countably many set operations (unions, intersections, and compliments). 
\subsection{Problem 3}
\par\vspace{0.3 cm}
Define a countably additive set function $\mu:\Sigma\rightarrow[0,1]$ with the property that $\mu(\emptyset)=0$ and 
$\mu(S)=1$.
\subsection{Problem 4}
\par\vspace{0.3 cm}
Let $E$ be the event that the coin first comes up heads on an even numbered trial.  Find $\mu(E)$.

\end{document}



