independent sigma algebras

Let $(\Omega,\mathcal{B},P)$ be a probability space. Let $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ be two sub sigma algebras of $\mathcal{B}$ . Then $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are said to be if for any pair of events $B_{1}\in\mathcal{B}_{1}$ and $B_{2}\in\mathcal{B}_{2}$ :

$P(B_{1}\cap B_{2})=P(B_{1})P(B_{2}).$

More generally, a finite set of sub- $\sigma$ -algebras $\mathcal{B}_{1},\ldots,\mathcal{B}_{n}$ is independent if for any set of events $B_{i}\in\mathcal{B}_{i}$ , $i=1,\ldots,n$ :

$P(B_{1}\cap\cdots\cap B_{n})=P(B_{1})\cdots P(B_{n}).$

An arbitrary set $\mathcal{S}$ of sub- $\sigma$ -algebras is mutually independent if any finite subset of $\mathcal{S}$ is independent.

The above definitions are generalizations of the notions of independence (http://planetmath.org/Independent) for events and for random variables:

1.

Events $B_{1},\ldots,B_{n}$ (in $\Omega$ ) are mutually independent if the sigma algebras $\sigma(B_{i}):=\{\varnothing,B_{i},\Omega-B_{i},\Omega\}$ are mutually independent.

2.

Random variables $X_{1},\ldots,X_{n}$ defined on $\Omega$ are mutually independent if the sigma algebras $\mathcal{B}_{X_{i}}$ generated by (http://planetmath.org/MathcalFMeasurableFunction) the $X_{i}$ ’s are mutually independent.

In general, mutual independence among events $B_{i}$ , random variables $X_{j}$ , and sigma algebras $\mathcal{B}_{k}$ means the mutual independence among $\sigma(B_{i})$ , $\mathcal{B}_{X_{j}}$ , and $\mathcal{B}_{k}$ .

Remark. Even when random variables $X_{1},\ldots,X_{n}$ are defined on different probability spaces $(\Omega_{i},\mathcal{B}_{i},P_{i})$ , we may form the product (http://planetmath.org/InfiniteProductMeasure) of these spaces $(\Omega,\mathcal{B},P)$ so that $X_{i}$ (by abuse of notation) are now defined on $\Omega$ 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 $\sigma$ -algebras
Defines	mutually independent sigma algebras