Two sigma-algebras are independent if, on a probability space (Ω,F,P), sub-sigma-algebras G1,G2⊆F satisfy
P(A∩B)=P(A)P(B)for all A∈G1,B∈G2.
A family (Gi)i∈I of sub-σ-algebras is independent if for every finite subset {i1,…,in}⊆I and every choice of events Ak∈Gik,
P(k=1⋂nAk)=k=1∏nP(Ak).
This formalizes “independence of information”: events determined by G1 do not influence probabilities of events determined by G2. In particular, independence of random variables X and Y can be characterized by independence of the generated σ-algebras σ(X) and σ(Y).