26.3. KOLMOGOROV EXTENSION THEOREM 719

Proposition 26.2.2 { fn} is Cauchy in probability, these being functions having valuesin a Banach space then there exists a set of measure zero N and a subsequence

{fnk

}such

that for ω /∈ N, limk→∞ fnk (ω) converges.

Proof: From the above definition, there exists n1 such that if m,n≥ n1, then

P(∥ fn− fm∥> 2−1)< 2−1

From the definition, there exists n2 > n1 such that P(∥ fn− fm∥> 2−2

)< 2−2 whenever

n,m≥ n1 and so forth. Thus

P(∥∥ fnk+1 − fnk

∥∥> 2−k)< 2−k

Letting Ak ≡[∥∥ fnk+1 − fnk

∥∥> 2−k], it follows from the Borell Cantelli lemma that there is

a set of measure zero, namely N ≡∩∞n=1∪k≥n Ak such that P

(NC)= 1= P

(∪∞

n=1∩k≥n ACk

).

To say ω ∈ NC is the same as saying that there exists n such that ω is in ACk for all k ≥ n.

In other words, eventually∥∥ fnk+1 − fnk

∥∥≤ 2−k. Now it follows that∥∥ fn+p (ω)− fnk (ω)∥∥≤ ∞

∑j=k

∥∥ fn+ j+1 (ω)− fn+ j (ω)∥∥< 2−(k−1)

if k is large enough. Hence{

fnk (ω)}

k is a Cauchy sequence for each ω /∈ N and since Eis complete, this sequence converges. ■

26.3 Kolmogorov Extension TheoremLet Mt be a complete separable metric space. This is called a Polish space. I will denotea totally ordered index set, (Like R) and the interest will be in building a measure on theproduct space, ∏t∈I Mt . If you like less generality, just think of Mt = Rkt or even Mt = R.By the well ordering principle, you can always put an order on any index set so this orderis no restriction, but we do not insist on a well order and in fact, index sets of great interestare R or [0,∞). Also for X a topological space, B (X) will denote the Borel sets.

Notation 26.3.1 The symbol J will denote a finite subset of I,J = (t1, · · · , tn) , the ti takenin order. EJ will denote a set which has a set Et of B (Mt) in the tth position for t ∈ J andfor t /∈ J, the set in the tth position will be Mt . KJ will denote a set which has a compact setin the tth position for t ∈ J and for t /∈ J, the set in the tth position will be Mt . Also denoteby RJ the sets EJ and R the union of all such RJ . Let EJ denote finite disjoint unions ofsets of RJ and let E denote finite disjoint unions of sets of R. Thus if F is a set of E , thereexists J such that F is a finite disjoint union of sets of RJ . For F ∈ Ω, denote by πJ (F )the set ∏t∈J Ft where F = ∏t∈I Ft .

With this preparation, here is the Kolmogorov extension theorem. It is Theorem 20.3.3proved earlier. In the statement and proof of the theorem, Fi,Gi, and Ei will denote Borelsets. Any list of indices from I will always be assumed to be taken in order. Thus, if J ⊆ Iand J = (t1, · · · , tn) , it will always be assumed t1 < t2 < · · ·< tn.

Theorem 26.3.2 For each finite set J = (t1, · · · , tn)⊆ I, suppose there exists a Borelprobability measure, νJ = ν t1···tn defined on the Borel sets of ∏t∈J Mt such that the followingconsistency condition holds. If (t1, · · · , tn)⊆ (s1, · · · ,sp) , then

ν t1···tn (Ft1 ×·· ·×Ftn) = νs1···sp

(Gs1 ×·· ·×Gsp

)(26.5)