\part{Probability Space}


\chapter{Sample spaces and events}

Considerj
\begin{definition}
A \emph{trial} is an experiment that produces a result, formally defined by a probability space (this comes later).
\end{definition}

\begin{definition}
An \emph{outcome} is a possible result of a trial. 
\end{definition}

\begin{definition}
An \emph{event} is a set of outcomes of a trial.
\end{definition}

\begin{definition}
The \emph{sample space} $\Omega$ is the set of all outcomes of a trial.
\end{definition}

\chapter{Probability}

Not all events are equal; events may differ in their 'probabilities'.

\begin{definition}
	The \emph{probability measure} $\mathrm{Pr} : \Omega \to [0,1]$ maps each event to its probability of occurence.
\end{definition}

One's understanding of Probability  One interpretation is that it describes the frequency of an event occuring in comparison to other events, given that the trial is conducted infinite amount of times; this is the \emph{frequentist philosophy} of probability theory.

The \emph{frequentist philosophy} interprets probability as a \emph{relative frequency}; the limiting ratio of how often an event occurs in comparison to other events as the amount of trials conducted approaches infinity. This interpretation is rather objective, and is inspired by the law of large numbers (which will be covered later).
The \emph{Bayesian philosophy} interprets probability as \emph{credence}; a level of condifence in comparsion to other events that the event will occur. This interpretation is rather subjective, and is inspired by Bayesian inference.


\begin{definition}
A pair of events are \emph{independent} iff the occurence of one event doesn't effect the probability of another.
\[ A \perp B \iff \mathrm{Pr}(A\cap B)=\mathrm{Pr}(A)\mathrm{Pr}(B)\]
\end{definition}


\begin{definition}
	A \emph{probability space} is a measure space of the 3-tuple $(\Omega, \mathcal{F}, \mathrm{Pr})$  such that
\begin{itemize}
	\item $\Omega$ is a sample space
	\item $\mathcal{F}$ is a $\sigma$-algebra over $\Omega$
	\item $\mathrm{Pr}: \mathal{F} \to [0,1]$ is the probability measure 
\end{itemize}
\begin{itemize}
	\item $\mathrm{Pr}(E)$ is a positive real number
	\item $\mathrm{Pr}(\Omega)=1$
	\item For disjoint $E_i$, one has $\mathrm{Pr}(\bigcup^{\infty}_{i=1}E_i) = \sum^{\infty}_{i=1} \mathrm{Pr}(E_i)$
\end{itemize}

\begin{definition}
An event occurs \emph{almost surely} iff the event has probability of 1. 
\[E \text{ occurs almost surely in } (\Omega, \mathcal{F},\mathrm{Pr}) \iff \mathrm{Pr}(E)=1 \]
\end{definition}





\chapter{Measure theoretic foundation}




- Cantelli's inequality
- Chebyshev's inequality
- Markov's inequality
- Filtration
- Filtered probability space

\begin{theorem}Left cosets are equivalence classes on $G$.
\end{theorem}

\begin{theorem}Left cosets are either equal or disjoint.
\end{theorem}

\chapter{Conditional probability}

- Conditional probability


\begin{theorem}\emph{Bayes' theorem}
Let $A,B$ be events, then the following relation holds.
\[\mathrm{Pr}(A|B)=\frac{\mathrm{Pr}(B|A)\mathrm{Pr}(A)}{\mathrm{Pr}(A)}\]
\[A,B \subseteq \Omega \implies \mathrm{Pr}(A|B)=\frac{\mathrm{Pr}(B|A)\mathrm{Pr}(A)}{\mathrm{Pr}(A)}\]
\end{theorem}





