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 independent 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 for events and for random variables:

1. Events $B_1,\ldots,B_n$ (in $\Omega$ are mutually independent if the sigma algebras $\sigma(B_i):=\lbrace \varnothing, B_i, \Omega-B_i, \Omega\rbrace$ 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 the $X_i$ s are mutually independent.

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 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.