Σ-algebra
Algebraic structure of set algebra
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Key Takeaways
- In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured.
- In probability theory, they are used to define events for which a probability can be defined.
- In formal terms, a σ-algebra (also σ-field , where the σ comes from the German "Summe", meaning "sum") on a set X {\displaystyle X} is a nonempty collection Σ {\displaystyle \Sigma } of subsets of X {\displaystyle X} closed under complement, countable unions, and countable intersections.
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with area or volume. In probability theory, they are used to define events for which a probability can be defined. 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 German "Summe", meaning "sum") on a set is a nonempty collection of subsets of closed under complement, countable unions, and countable intersections. The ordered pair