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Sigma-algebras G₁, G₂, ... are independent if P(A₁ ∩ A₂ ∩ ...) = P(A₁)P(A₂)... for all Aᵢ ∈ Gᵢ. Random variables are independent if the sigma-algebras they generate are independent. This definition extends to countably many sigma-algebras, unifying discrete and continuous independence.
From your study of probability spaces, you know that a probability space (Ω, ℱ, P) consists of a sample space, a sigma-algebra of events, and a probability measure. The sigma-algebra ℱ captures *all the events we can talk about*. Independence of sigma-algebras extends the elementary notion — that P(A ∩ B) = P(A) · P(B) for independent events — from individual events to entire collections of them. Two sigma-algebras G₁ and G₂ are independent if the factorization P(A₁ ∩ A₂) = P(A₁) · P(A₂) holds for every A₁ ∈ G₁ and every A₂ ∈ G₂. Knowing something about any event in G₁ tells you nothing about any event in G₂.
The definition connects directly to random variables. The sigma-algebra generated by a random variable X, written σ(X), is the collection of all events of the form {ω : X(ω) ∈ B} for Borel sets B. Roughly, σ(X) captures all information that X could possibly reveal about the underlying outcome ω. Two random variables X and Y are independent (in the rigorous measure-theoretic sense) if and only if σ(X) and σ(Y) are independent sigma-algebras. For discrete random variables this reduces to the familiar P(X = x, Y = y) = P(X = x) · P(Y = y); for continuous ones it requires the joint density to factor as f_{X,Y}(x,y) = f_X(x) · f_Y(y). The sigma-algebra definition unifies both cases without special-casing.
The extension to infinite collections is where the measure-theoretic framework pays off. A countable sequence of sigma-algebras G₁, G₂, G₃, ... is mutually independent if for every finite subcollection and every choice of one event from each, the probability of their intersection equals the product of the individual probabilities. This is the right definition for a sequence of random variables X₁, X₂, ... to be i.i.d. (independent and identically distributed). The "identically distributed" part is a condition on marginal distributions; the "independent" part is exactly the statement that σ(X₁), σ(X₂), ... are mutually independent.
This definition is the foundation for the laws of large numbers and the central limit theorem at the rigorous level. Both theorems concern sums of independent random variables, and their proofs require the full machinery of sigma-algebra independence — particularly tools like the Borel-Cantelli lemma and characteristic functions, which are stated in terms of sigma-algebras. A key subtlety: pairwise independence of G₁, ..., Gₙ does not imply mutual independence. Mutual independence requires the product rule for every subset, not just every pair. This distinction matters for some CLT extensions and is one of the reasons the measure-theoretic definition is stated the way it is.