D - Kolmogorov extension theorem

Summary

Suppose S is a metric space.We use S^ℕ for the product space S×S×… furnished with the product topology. We may view S^ℕ as the set of sequences (x₁, x₂, …) of elements of S. We use the σ-field on S^ℕ generated by the cylindrical sets. Given an element x = (x₁, x₂, …) of S^ℕ, we define π_n(x) = (x₁, …, x_n) ∈ _Sn.

We suppose we have a Radon probability measure μ_n defined on S_n for each n. (Being a Radon measure means that we can approximate μ_n(A) from below by compact sets; see

Folland (1999) for details.) The μ_n are consistent if μ_n+1(A × S) = μ_n(A) whenever A is a Borel subset of Sⁿ. The Kolmogorov extension theorem is the following.

Theorem D.1Suppose for each n we have a probability measure μ_non Sⁿ. Suppose the μ_n's are consistent. Then there exists a probability measure μ S^ℕ such that μ(A × S^ℕ) = μ_n(A) for all A ⊂ Sⁿ.

Proof Define μ on cylindrical sets by μ(A × S^ℕ) = μ_n(A) if A ⊂ Sⁿ. By the consistency assumption, μ is well defined. By the Carathéodory extension theorem, we can extend μ to the σ-field generated by the cylindrical sets provided we show that whenever A_n are cylindrical sets decreasing to ∅, then μ(A_n) → 0.

Suppose A_n are cylindrical sets decreasing to ∅ but μ(A_n) does not tend to 0; by taking a subsequence we may assume without loss of generality that there exists ε > 0 such that μ(A_n) ≥ ε for all n. We will obtain a contradiction.