independent sigma algebras

Let $(\mathrm{\Omega },ℬ,P)$ be a probability space. Let ${ℬ}_1$ and ${ℬ}_2$ be two sub sigma algebras of $ℬ$. Then ${ℬ}_1$ and ${ℬ}_2$ are said to be if for any pair of events $B_1\in {ℬ}_1$ and $B_2\in {ℬ}_2$:

$$P(B_1\cap B_2)=P(B_1)P(B_2).$$

More generally, a finite set of sub-$\sigma$-algebras ${ℬ}_1,\mathrm{\dots },{ℬ}_n$ is independent if for any set of events $B_i\in {ℬ}_i$, $i=1,\mathrm{\dots },n$:

$$P(B_1\cap \mathrm{\cdots }\cap B_n)=P(B_1)\mathrm{\cdots }P(B_n).$$

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

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

1. Events $B_1,\mathrm{\dots },B_n$ (in $\mathrm{\Omega }$) are mutually independent if the sigma algebras $\sigma (B_i):=\{\mathrm{\varnothing },B_i,\mathrm{\Omega }-B_i,\mathrm{\Omega }\}$ are mutually independent.

2. Random variables $X_1,\mathrm{\dots },X_n$ defined on $\mathrm{\Omega }$ are mutually independent if the sigma algebras ${ℬ}_{X_i}$ 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 ${ℬ}_k$ means the mutual independence among $\sigma (B_i)$, ${ℬ}_{X_j}$, and ${ℬ}_k$.

Remark. Even when random variables $X_1,\mathrm{\dots },X_n$ are defined on different probability spaces $({\mathrm{\Omega }}_i,{ℬ}_i,P_i)$, we may form the product (http://planetmath.org/InfiniteProductMeasure) of these spaces $(\mathrm{\Omega },ℬ,P)$ so that $X_i$ (by abuse of notation) are now defined on $\mathrm{\Omega }$ and their independence can be discussed.