\part{Advanced}

\chapter{Probability space}

Measure theory is an extremely powerful characterization for probability theory, and we can form probability theory as a special type of measure space called a \emph{probability space}




- Cantelli's inequality
- Chebyshev's inequality
- Markov's inequality
- Filtration
- Filtered probability space
- Expectation
\begin{definition}[Expectation]
\[ \mathrm{E} [ X ] = \int_{\Omega} X d\mathrm{Pr} \]
\end{definition}


\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}

- Convergence in distribution
- Convergence in probability
- Almost sure convergence