# 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. 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. 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 IndependentSigmaAlgebras 2013-03-22 16:22:58 2013-03-22 16:22:58 CWoo (3771) CWoo (3771) 9 CWoo (3771) Definition msc 60A05 mutually independent $\sigma$-algebras mutually independent sigma algebras