*Taken from Real Analysis and Probability by R.M. Dudley.*

For a sequence of repeated, independent trials of an experiment, some probability distributions and variables converge as tends to infinity. In proving such limit theorems, it is useful to be able to construct a probability space on which a sequence of independent random variables is defined in a natural way; specifically, as coordinates for a countable Cartesian product.

The Cartesian product of finitely many -finite measure spaces gives a -finite measure space. For example, Cartesian products of Lesbesgue measure on the line give Lesbesgue measure on finite-dimensional Euclidean spaces. But suppose we take a measure space with two points each having measure , , and form a countable Cartesian product of copies of this space, so that the measure of any countable product of sets equals the product of their measures. Then we would get an uncountable space in which all singletons have measure , giving the measure usually called *counting measure*. An uncountable set with counting measure is not a -finite space, although in this example it was a countable product of finite measure spaces. By contrast, the the countable product of probability measures *will *again be a probability space. Here are some definitions.

For each let be a probability space. Let be the Cartesian product , that is, the set of all sequences with for all . Let be the natural projection of onto for each : for all . Let be the smallest -algebra of subsets of such that for all , is measurable from to . In other words, is the smallest -algebra containing all sets for all and all .

Let be the collection of all sets where for all and except for at most finitely many values of . Elements of will be called *rectangles*. Now recall the notion of semiring. has this property.

### Definition

For any set , a collection is called a *semiring *if and for any and in , we have and for some finite and disjoint .

## Proposition

*The collection if rectangles in the infinite product is a semiring. The algebra generated by is the collection of finite disjoint unions of elements of .*

*Proof *: If and are any two rectangles, then clearly is a rectangle (). In a product of two spaces, the collection of rectangles is a semiring (p.95, Proposition 3.2.2). Specifically, a difference of two rectangles is a union of two disjoint rectangles:

.

It follows by induction that in any finite Cartesian product, any difference of rectangles is a finite disjoint union of rectangles. Thus is a semiring. We have , so the ring generated by is an algebra (p.96, Proposition 3.2.3). Since every algebra is a ring, is the algebra generated by . By Proposition 3.2.3, consists of all finite disjoint unions of elements of

Now for , let . The product converges since all but finitely many factors are . Here is the main theorem to be proved in the rest of this section:

## Theorem: Existence Theorem for Infinite Product Probabilities

* on extends uniquely to a (countably additive) probability measure on .*

*Proof *: For each , write as a finite disjoint union of sets in , say

,

and define

.

Let us first show that is well-defined and finitely additive on finite disjoint unions. Each is a product of sets with for all for some finite . Let be the maximum of the for . Then since all the equal for , properties of on such sets are equivalent to properties of the finite product measure on . To show that is well-defined, if a set of two different finite disjoint union of sets in , we can take the maximum of the values of for the two unions and still have a finite product. So is well-defined and finitely additive on by the finite product measure theorem (p.139 , Theorem 4.4.6).

If us countably additive on , then it has a unique countably additive extension to by the Carathéodory Extension Theorem. So it’s enough to prove countable additivity on . Equivalently, if , , and we want to prove (“ is continuous at ” — p.86, Theorem 3.1.1). In other words, if is a decreasing sequence of sets in and for some , for all , we must show .

Let on . For each , let

.

Let and be defined on just as and were on . For each and , , let

.

For a set in a product space and , let . If is in a product -algebra then (due to the proof of Theorem 4.4.3, p. 135). For any there is an large enough so that

for some .

Since is a finite union of rectangles with this property, take the maximum of the values of for the rectangles. Then

where

for some , , . Now for any , and , , where is the union of those sets

such that for all .

Thus , so of it is defined. Then by the Tonelli-Fubini theorem in

,

we have

.

For with for all , let

.

For each , apply the formula for to , . Then

.

Thus for all . As increases, the sets decrease; thus, so do the and the .

Since is countably additive,

by monotone convergence of indicator functions, so . Take any . Let and . Then decreases as increases,

for al ,

and for all , so the intersection of all the is non-empty in and we can choose in it.

Inductively, by the same argument there are for all such that for all and . Let . To prove that for each , choose large enough (depending on ) so that for all , or . This is possible since . Then , so . Hence

Actually, this theorem holds for arbitrary (not necessarily countable) products of probability spaces. The proof needs no major change, since each set in the -algebra depends only on countably many coordinates. In other words, given a product , where is a possibly uncountable index set, for each set there is a countable subset of and a set such that .

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July 5, 2012 at 7:44 pm

Infinite product probability « unmappingproperty[…] post I’d like to introduce the notion of infinite product of probability spaces, following this. This result (their existence, uniqueness is standard) is attributed, as far as I know, to Ulam, […]