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\title[My Project]{Probability and Measure Theory}
\author[\tiny{Madeleine Lane Margot Kandarian}]{Madeleine Lane and Margot Kandarian\\ \scriptsize{\url{sites.google.com/site/probabilityandmeasuretheory}}}
\institute[USF]{University of San Francisco\\San Francisco, CA}
\date[\today ]{Presentation at USF Math 235}
\maketitle
\begin{frame}\frametitle{Introduction}
\begin{block}{Definition.}
Probability is how \textit{likely} something is to happen. \pause
\end{block}
$$P(A)$$\pause
$A$ is a set that stands for an event and $P$ stands for the probability of that event occurring. \pause
\begin{block}{Definition.}
A measure $\mu$ on $\Omega$ with $\sigma$-algebra $A$ is a function $\mu : A \rightarrow [0, \infty]$. \pause
\end{block}
A measure is a generalization of the concepts of length, area, and volume.
\end{frame}
\begin{frame}\frametitle{Notation.}
\begin{itemize}
\item All of the possible outcomes of an experiment is denoted by the sample space, $\Omega$.
\item $\Omega$ is a sure event and $\emptyset$, the empty set, is an impossible event.
\item An event $A$ is a set of outcomes
\item $P(\Omega) = 1$ and $P(\emptyset)=0$
\item $P(A)$ is a nonnegative number and $0 \leq P(A) \leq 1$
\item A measurable space is written as $(\Omega, A)$
\item $A$ is a $\sigma$-algebra, meaning it is "measurable" on some space
\end{itemize}
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\begin{frame}\frametitle{$\mathbb{Q}$ has a Lebesgue measure of 0.}
\begin{proof}
The rational numbers, $\mathbb{Q}$, are countably infinite, so we can say that $Q = \bigcup\limits_{i=1}^\infty {q_i}$. Then $\mu(Q)$ = $\sum\limits_{i=1}^\infty {\mu(q_i)}$. Since the measure of any set of cardinality 1 (each point, $q_i$, representing a rational number) has measure of 0, $$\sum\limits_{i=1}^\infty {\mu(q_i)} = \sum\limits_{i=1}^\infty 0 = 0$$. Therefore, the rational numbers have a Lebesgue measure of 0.
\end{proof}
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\begin{frame}\frametitle{Why probability and measure theory?}
\begin{itemize}
\item A mathematical foundation for statistics \pause
\item Relevant to statistical mechanics and quantum computing \pause
\item Everyday life: predicting weather, coin tosses at sporting events \pause
\item Probability theory on its own fails when considering events with infinitely many outcomes \pause
\item In measure theory, it is possible to speak of the area, or measure, of a square cut up into infinite pieces and spread out over an infinite plane
\end{itemize}
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\begin{frame}\frametitle{A measure vs. a probability measure.}
A measure of a set $A$ is written as $$ m(A)=\sum_{n} m(A_{n}) $$ \pause
Now suppose we have a set of $n$ nonnegative numbers $\left\{p_{1}, p_{2},...,p_{n}\right\}$ so that $\sum\limits_{i=1}^n p_{i}=1$. \pause
\vspace{.5cm}
If the sample space is finite, meaning $\Omega = \left\{\omega_{1},...,\omega_{n}\right\}$, then there exists a unique probability distribution $P$ over events of $\Omega$ such that $P(\left\{\omega_{i}\right\}) = p_{i}$. If $E = \left\{\omega_{i}
: i \in I\right\}$ then the probability measure is written as
$$P(E) = \sum_{i\in I} P(\left\{\omega_{i}\right\}) = \sum_{i \in I} p_{i}.$$
\end{frame}
\begin{frame}{Example of a probability measure.}
$$P(E) = \sum_{i\in I} P(\left\{\omega_{i}\right\}) = \sum_{i \in I} p_{i}.$$ \pause
Suppose you throw two distinct dice where the sample space is $\Omega =\left\{(x,y):x,y\in \left\{1,2,...,6\right\}\right\}.$ The probability of the event
$E =\left\{(4, 4)\right\}$ is therefore $P(E) = \frac{1}{|\Omega|}= \frac{1}{36}.$ \pause \vspace{.5cm}
The probability of getting a sum of 9 in both dice is $$P(sum = 9) = P(\left\{(6,3),(3,6),(4,5),(5,4)\right\})$$ \center{which equals}
$$ \frac{|\left\{(6, 3),(3, 6),(4, 5),(5, 4)\right\}|}{36}=\frac{4}{36}.$$
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\begin{frame}{THANK YOU!}
References
\begin{itemize}
\item http://mathworld.wolfram.com/Measure.html
\item "Probability Theory". \textit{Wikipedia}.
\item Lebanon, Guy. "Examples of Probability Measures".
\item http://austinrochford.com/posts/2013-12-31-almost-no-rationals.html
\end{itemize}
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