Independence of sigma-algebras

Two sigma-algebras are independent if, on a probability space $(\Omega,\mathcal F,\mathbb P)$, sub-sigma-algebras $\mathcal G_1,\mathcal G_2\subseteq\mathcal F$ satisfy

A family $(\mathcal G_i)_{i\in I}$ of sub-$\sigma$-algebras is independent if for every finite subset $\{i_1,\dots,i_n\}\subseteq I$ and every choice of events $A_k\in \mathcal G_{i_k}$,

This formalizes “independence of information”: events determined by $\mathcal G_1$ do not influence probabilities of events determined by $\mathcal G_2$. In particular, independence of random variables $X$ and $Y$ can be characterized by independence of the generated $\sigma$-algebras $\sigma(X)$ and $\sigma(Y)$.

Examples:

• In two independent coin flips, let $\mathcal G_1$ be the $\sigma$-algebra generated by the first flip and $\mathcal G_2$ the $\sigma$-algebra generated by the second flip. Then $\mathcal G_1$ and $\mathcal G_2$ are independent.
• If $\mathcal G_2\subseteq \mathcal G_1$ and $\mathcal G_2$ contains an event $B$ with $\mathbb P(B)\in(0,1)$, then $\mathcal G_1$ and $\mathcal G_2$ are not independent (since taking $A=B$ violates $\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)$).