Editing Probability axioms (section)

== Kolmogorov axioms ==
The assumptions as to setting up the axioms can be summarised as follows: Let <math>(\Omega, F, P)</math> be a [[measure space]] such that <math>P(E)</math> is the [[probability]] of some [[Event (probability theory)|event]] <math>E</math>, and <math>P(\Omega) = 1</math>. Then <math>(\Omega, F, P)</math> is a [[probability space]], with sample space <math>\Omega</math>, event space <math>F</math> and [[probability measure]] <math>P</math>.<ref name=":0" />

==={{Anchor|Non-negativity}}First axiom ===
The probability of an event is a non-negative real number:
:<math>P(E)\in\mathbb{R}, P(E)\geq 0 \qquad \forall E \in F</math>

where <math>F</math> is the event space. It follows (when combined with the second axiom) that <math>P(E)</math> is always finite, in contrast with more general [[Measure (mathematics)|measure theory]]. Theories which assign [[negative probability]] relax the first axiom.

=== {{Anchor|Unitarity|Normalization}}Second axiom ===
This is the assumption of [[unit measure]]: that the probability that at least one of the [[elementary event]]s in the entire sample space will occur is 1.

: <math>P(\Omega) = 1</math>

=== {{Anchor|Sigma additivity|Finite additivity|Countable additivity|Finitely additive}}Third axiom ===
This is the assumption of [[σ-additivity]]:
: Any [[countable]] sequence of [[disjoint sets]] (synonymous with ''[[Mutual exclusivity|mutually exclusive]]'' events) <math>E_1, E_2, \ldots</math> satisfies
::<math>P\left(\bigcup_{i = 1}^\infty E_i\right) = \sum_{i=1}^\infty P(E_i).</math>
Some authors consider merely [[finitely additive]] probability spaces, in which case one just needs an [[field of sets|algebra of sets]], rather than a [[σ-algebra]].<ref>{{Cite web|url=https://plato.stanford.edu/entries/probability-interpret/#KolProCal|title=Interpretations of Probability|last=Hájek|first=Alan|date=August 28, 2019|website=Stanford Encyclopedia of Philosophy|access-date=November 17, 2019}}</ref> [[Quasiprobability distribution]]s in general relax the third axiom.