Chapter 7 Infinite product spaces

7.4. KOMOGOROV’S EXTENSION THEOREM 11
7.4 Komogorov’s extension theorem
We now address the corresponding extension of measures to infinite product spaces:
Theorem 7.16 [Kolmogorov’s extension theorem] Let M be a compact metric space,
B = B(M) the Borel s-algebra on M and, for each n 1, let µn be a probability measure
on (M n , B ⌦n ). We suppose these measures are consistent in the following sense:
8n 1, 8A1,...,An 2 B : µn+1(A1 ⇥···⇥An ⇥ M) =µn(A1 ⇥···⇥An).
Then there exists a unique probability measure µ on (M N , B ⌦N ) such that
µ(A1 ⇥···⇥An ⇥ R ⇥ R ...)=µn(A1 ⇥···⇥An) (7.26)
holds for any n 1 and any Borel sets Ai 2R.
As many extension arguments from measure theory, the proof boils down to proving
that if a sequence of sets decreases to an empty set, then the measures decrease
to zero. In our case this will be ensure through the following lemma:
Lemma 7.17 Let W be a metric space and let A be an algebra of subsets of W (which is
not necessarily a s-algebra). Suppose that µ : A ! [0, 1] is a finitely-additive set function
which is regular in the following sense: For each set B 2 A and each e > 0, there exists a
compact set C 2 A such that
C ⇢ B (7.27)
and
µ(B \ C) apple e. (7.28)
Then for each decreasing sequence Bn 2 A with Bn # ∆ we have µ(Bn) # 0.
Proof. Let Bn be a decreasing sequence of sets with limn!• µ(Bn) =d > 0. We will
show that T
n Bn 6= ∆. By the assumptions, we can find compact sets Cn 2 A such
that Cn ⇢ Bn and µ(Bn \ Cn) < d2 n 1 . Now since Bn Bn+1, we have
Bn \
n[
(Bk \ Ck) ⇢
k=1
n\
Ck. Indeed, if x 2 Bn then x 2 B k for all k apple n and then x can only belong to the set on
the left if x 2 C k for all k apple n. Using finite additivity of µ, it follows that
⇣ \ n
µ
k=1
C k
⌘
µ(Bn)
k=1
n
 µ(Bk \ Ck) d  µ(Bk \ Ck) k=1
k 1
Consequently, since finite additivity implies that µ(∆) =0, the intersection T n k=1 C k
is non-empty for any finite n. But the sets Cn are compact and thus T • k=1 C k 6= ∆.
The inclusions Cn ⇢ Bn imply that also T
n Bn 6= ∆.
Lemma 7.17 links measure with topology (which is the reason why we often consider
standard Borel spaces). Next we observe:
d
2 .