sigma-algebra
Sign in to saveAlso known as σ-algebra, sigma-field, σ-field
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.
~23 min read
Article
28 sectionsContents
- Examples of σ-algebras
- Motivation
- Measure
- Limits of sets
- Sub σ-algebras
- Definition and properties
- Definition
- Dynkin's π-λ theorem
- Combining σ-algebras
- σ-algebras for subspaces
- Relation to σ-ring
- Typographic note
- Particular cases and examples
- Separable σ-algebras
- Simple set-based examples
- Stopping time sigma-algebras
- σ-algebras generated by families of sets
- σ-algebra generated by an arbitrary family
- σ-algebra generated by a function
- Borel and Lebesgue σ-algebras
- Product σ-algebra
- σ-algebra generated by cylinder sets
- Ball σ-algebra
- σ-algebra generated by random variable or vector
- σ-algebra generated by a stochastic process
- See also
- References
- External links
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 X is a nonempty collection \Sigma of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair (X, \Sigma) is called a measurable space.