Independent sigma-algebras

Two sub-sigma-algebras ${\mathcal{F}}_1,{\mathcal{F}}_2$ are independent whenfor every $A\in {\mathcal{F}}_1$ and $B\in {\mathcal{F}}_2$.

Events $A_1,\dots ,A_n$ are mutually independent whenfor every finite index set $J$.

If two pi-systems satisfy the probability factorization pairwise, their generated sigma-algebras are independent. For one event in the first family, the events in the second sigma-algebra satisfying the factorization form a Dynkin system; apply Dynkin lemma, then repeat with the roles reversed.

For a sequence of random variables $(X_n)$, the tail sigma-algebra isIts events are unaffected by changing any finite initial segment.

For a sequence of finite real random variables, ${\mathrm{limsup}}_n{n}^{-1}{\sum }_{j=1}^n{X}_j$ is measurable with respect to its tail sigma-algebra: deleting finitely many summands changes each sample mean by a quantity tending to zero. When the random variables are independent random variables, the Kolmogorov zero-one law makes every finite such limit constant almost surely.

A tail event is measurable with respect to a tail sigma-algebra, so changing finitely many coordinates does not affect whether it occurs.

The tail sigma-algebra of a sequence of independent random variables is trivial: every tail event has probability zero or one.