Sigma algebra

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In mathematics, a sigma algebra is a formal mathematical structure intended among other things to provide a rigid basis for measure theory and axiomatic probability theory. In essence it is a collection of subsets of an arbitrary set that contains itself and which is closed under the taking of complements (with respect to ) and countable unions. It is found to be just the right structure that allows construction of non-trivial and useful measures on which a rich theory of (Lebesgue) integration can be developed which is much more general than Riemann integration.

Formal definition

Given a set , let be its power set, i.e. set of all subsets of . Then a subset FP (i.e., F is a collection of subset of ) is a sigma algebra if it satisfies all the following conditions or axioms:

  1. If then the complement
  2. If for then

Examples

  • For any set S, the power set 2S itself is a σ algebra.
  • The set of all Borel subsets of the real line is a sigma-algebra.
  • Given the set = {Red, Yellow, Green}, the subset F = {{}, {Green}, {Red, Yellow}, {Red, Yellow, Green}} of is a σ algebra.

See also

Set

Set theory

Borel set

Measure theory

Measure

External links

  • Tutorial on sigma algebra at probability.net