σ-algebra

In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a> and in <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, a σ-algebra ("sigma algebra") is part of the formalism for defining <a href="/facts/Measurable_set/yCq7zlI4">sets that can be measured</a>. In <a href="/facts/Calculus/MEmUTYoz">calculus</a> and <a href="/facts/Mathematical_analysis/XXeR26Ih">analysis</a>, for example, σ-algebras are used to define the concept of sets with <a href="/facts/Area/KsNG1G9t">area</a> or <a href="/facts/Volume/aeAXCBUd">volume</a>. In probability theory, they are used to define events with a well-defined probability. In this way, σ-algebras help to formalize the notion of size.
In formal terms, a σ-algebra (also σ-field, where the σ comes from the <a href="/facts/German_language/oWaQIDA8">German</a> "Summe", meaning "sum") on a set X is a nonempty collection Σ of <a href="/facts/Subsets/SJGERmbA">subsets</a> of X <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closed</a> under <a href="/facts/Complement_(set_theory)/yWUePIYY">complement</a>, countable <a href="/facts/Union_(set_theory)/KVVXKfWl">unions</a>, and countable <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersections</a>. The ordered pair 
 
 
 
 (
 X
 ,
 Σ
 )
 
 
 {\displaystyle (X,\Sigma )}
 
 is called a <a href="/facts/Measurable_space/AJ98lhT4">measurable space</a>. 
The set X is understood to be an ambient space (such as the 2D plane or the set of outcomes when rolling a six-sided die {1,2,3,4,5,6}), and the collection Σ is a choice of subsets declared to have a well-defined size. The closure requirements for σ-algebras are designed to capture our intuitive ideas about how sizes combine: if there is a well-defined probability that an event occurs, there should be a well-defined probability that it does not occur (closure under complements); if several sets have a well-defined size, so should their combination (countable unions); if several events have a well-defined probability of occurring, so should the event where they all occur simultaneously (countable intersections).
The definition of σ-algebra resembles other mathematical structures such as a <a href="/facts/Topological_space/v2ut7CbK">topology</a> (which is required to be closed under all unions but only finite intersections, and which doesn't necessarily contain all complements of its sets) or a <a href="/facts/Set_algebra/4E3vQl4f">set algebra</a> (which is closed only under finite unions and intersections).

σ-algebra open-in-new

σ-algebra