1864 CHAPTER 59. BASIC PROBABILITY

Now to conclude the proof, apply the Caratheodory extension theorem to obtain Pa probability measure which extends P0 to a σ algebra which contains σ (E ) the sigmaalgebra generated by E with P = P0 on E . Thus for EJ ∈ E , P(EJ) = P0 (EJ) = νJ (PJE j) .

Next, let (∏t∈I Mt ,F ,P) be the probability space and for x ∈ ∏t∈I Mt let Xt (x) = xt ,the tth entry of x. It follows Xt is measurable (also continuous) because if U is open in Mt ,then X−1

t (U) has a U in the tth slot and Ms everywhere else for s ̸= t. Thus inverse imagesof open sets are measurable. Also, letting J be a finite subset of I and for J = (t1, · · · , tn) ,and Ft1 , · · · ,Ftn Borel sets in Mt1 · · ·Mtn respectively, it follows FJ , where FJ has Fti in thetthi entry, is in E and therefore,

P([Xt1 ∈ Ft1 ]∩ [Xt2 ∈ Ft2 ]∩·· ·∩ [Xtn ∈ Ftn ]) =

P([(Xt1 ,Xt2 , · · · ,Xtn) ∈ Ft1 ×·· ·×Ftn ]) = P(FJ) = P0 (FJ)

= ν t1···tn (Ft1 ×·· ·×Ftn)

Finally consider the claim about the integrals. Suppose f (xt1 , · · · ,xtn) = XF where Fis a Borel set of ∏t∈J Mt where J = (t1, · · · , tn). To begin with suppose

F = Ft1 ×·· ·×Ftn (59.2.8)

where each Ft j is in B(Mt j

). Then∫

Mt1×···×Mtn

XF (xt1 , · · · ,xtn)dν t1···tn = ν t1···tn (Ft1 ×·· ·×Ftn)

= P

(∏t∈I

Ft

)=∫

Ω

X∏t∈I Ft (x)dP

=∫

Ω

XF (xt1 , · · · ,xtn)dP (59.2.9)

where Ft = Mt if t /∈ J. Let K denote sets, F of the sort in 59.2.8. It is clearly a π system.Now let G denote those sets F in B (∏t∈J Mt) such that 59.2.9 holds. Thus G ⊇K . It isclear that G is closed with respect to countable disjoint unions and complements. HenceG ⊇ σ (K ) but σ (K ) = B (∏t∈J Mt) because every open set in ∏t∈J Mt is the countableunion of rectangles like 59.2.8 in which each Fti is open. Therefore, 59.2.9 holds for everyF ∈B (∏t∈J Mt) .

Passing to simple functions and then using the monotone convergence theorem yieldsthe final claim of the theorem.

59.3 IndependenceThe concept of independence is probably the main idea which separates probability fromanalysis and causes some of us to struggle to understand what is going on.