View source for Measure-theoretic Probability Theory

Graduate-level Probability Theory

==Probability Measures==
===Definitions===
;Field
A class <math>\mathcal{F}</math> of subsets of <math>\Sigma</math> is called a field if 
it contains <math>\Sigma</math> and is closed under complements and finite unions.
* <math>\Sigma \in F</math>
* <math>A \in F \implies A^c \in F</math>
* <math> A, B \in F \implies A \cup B \in F</math>

;Sigma-field
;A class F is called a <math>\sigma</math>-field if it is also closed under countable unions
* <math>A_1, A_2, ... \in F \implies A_1 \cup A_2 \cup ... \in F</math>

;Probability measure space
Given a sigma field F on space <math>\Sigma</math> and probability measure P,<br>
<math>(\Sigma, F, P)</math> is a probability measure space.


==Textbooks==
* [Probability and Measure by Patrick Billingsley]