Editing Probability space (section)

== Definition ==
In short, a probability space is a [[measure space]] such that the measure of the whole space is equal to one.

The expanded definition is the following: a probability space is a triple <math>(\Omega,\mathcal{F},P)</math> consisting of:
* the [[sample space]] <math>\Omega</math> – an arbitrary [[non-empty set]],
* the [[σ-algebra]] <math>\mathcal{F} \subseteq 2^\Omega</math> (also called σ-field) – a set of subsets of <math>\Omega</math>, called [[event (probability theory)|events]], such that:
** <math>\mathcal{F}</math> contains the sample space: <math>\Omega \in \mathcal{F}</math>,
** <math>\mathcal{F}</math> is closed under [[complement (set theory)|complements]]: if <math>A\in\mathcal{F}</math>, then also <math>(\Omega\setminus A)\in\mathcal{F}</math>,
** <math>\mathcal{F}</math> is closed under [[countable set|countable]] [[Union (set theory)|unions]]: if <math>A_i\in\mathcal{F}</math> for <math>i=1,2,\dots</math>, then also <math display="inline"> (\bigcup_{i=1}^\infty A_i)\in\mathcal{F}</math>
*** The corollary from the previous two properties and [[De Morgan's law]] is that <math>\mathcal{F}</math> is also closed under countable [[Intersection (set theory)|intersections]]: if <math>A_i\in\mathcal{F}</math> for <math>i = 1,2,\dots</math>, then also <math display="inline"> (\bigcap_{i=1}^\infty A_i)\in\mathcal{F}</math>
* the [[probability measure]] <math>P:\mathcal{F}\to[0,1]</math> – a function on <math>\mathcal{F}</math> such that:
** ''P'' is [[countably additive]] (also called σ-additive): if <math>\{A_i\}_{i=1}^\infty \subseteq \mathcal{F}</math> is a countable collection of pairwise [[disjoint sets]], then <math display="inline"> P(\bigcup_{i=1}^\infty A_i)=\sum_{i=1}^\infty P(A_i),</math>
** the measure of the entire sample space is equal to one: <math>P(\Omega)=1</math>.