Editing Field of sets (section)

=== Sigma algebras and measure spaces ===

If an algebra over a set is closed under countable [[Union (set theory)|unions]] (hence also under [[countable]] [[Intersection (set theory)|intersections]]), it is called a [[sigma algebra]] and the corresponding field of sets is called a '''measurable space'''. The complexes of a measurable space are called '''measurable sets'''. The [[Lynn Harold Loomis|Loomis]]-[[Roman Sikorski|Sikorski]] theorem provides a Stone-type duality between countably complete Boolean algebras (which may be called '''abstract sigma algebras''') and measurable spaces.

A '''measure space''' is a triple <math>( X, \mathcal{F}, \mu  )</math> where <math>( X, \mathcal{F} )</math> is a measurable space and <math>\mu</math> is a [[Measure theory|measure]] defined on it. If <math>\mu</math> is in fact a [[Probability theory|probability measure]] we speak of a '''probability space''' and call its underlying measurable space a '''sample space'''. The points of a sample space are called '''sample points''' and represent potential outcomes while the measurable sets (complexes) are called '''events''' and represent properties of outcomes for which we wish to assign probabilities. (Many use the term '''sample space''' simply for the underlying set of a probability space, particularly in the case where every subset is an event.) Measure spaces and probability spaces play a foundational role in [[measure theory]] and [[probability theory]] respectively.

In applications to [[Physics]] we often deal with measure spaces and probability spaces derived from rich mathematical structures such as [[inner product space]]s or [[topological group]]s which already have a topology associated with them - this should not be confused with the topology generated by taking arbitrary unions of complexes.