independent sigma algebras

Let (Ω,,P) be a probability spaceMathworldPlanetmath. Let 1 and 2 be two sub sigma algebras of . Then 1 and 2 are said to be if for any pair of events B11 and B22:


More generally, a finite setMathworldPlanetmath of sub-σ-algebras 1,,n is independent if for any set of events Bii, i=1,,n:


An arbitrary set 𝒮 of sub-σ-algebras is mutually independent if any finite subset of 𝒮 is independent.

The above definitions are generalizationsPlanetmathPlanetmath of the notions of independence ( for events and for random variablesMathworldPlanetmath:

  1. 1.

    Events B1,,Bn (in Ω) are mutually independent if the sigma algebras σ(Bi):={,Bi,Ω-Bi,Ω} are mutually independent.

  2. 2.

    Random variables X1,,Xn defined on Ω are mutually independent if the sigma algebras Xi generated by ( the Xi’s are mutually independent.

In general, mutual independence among events Bi, random variables Xj, and sigma algebras k means the mutual independence among σ(Bi), Xj, and k.

Remark. Even when random variables X1,,Xn are defined on different probability spaces (Ωi,i,Pi), we may form the productPlanetmathPlanetmath ( of these spaces (Ω,,P) so that Xi (by abuse of notation) are now defined on Ω and their independence can be discussed.

Title independent sigma algebras
Canonical name IndependentSigmaAlgebras
Date of creation 2013-03-22 16:22:58
Last modified on 2013-03-22 16:22:58
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 60A05
Synonym mutually independent σ-algebras
Defines mutually independent sigma algebras