ABSTRACT
This chapter describes infinite dimensional product measures spaces, where infinite meants countably infinite. Thus the finite products of measure spaces are generalized to countable products. In the same way that points in a finite product space can be identified with n -tuples of variates, points in countably infinite product spaces are identified with sequences of variates. Having infinitely many dimensions raises definitional issues almost immediately for the definition of measurable rectangles. There are also technical difficulties in the proof of countable additivity on an algebra, and for this one may require a more general algebra formulation than that provided by the measurable rectangles. For more generality, these probability spaces must be assumed to be complete metric spaces, and the associated probability measures must then be assumed to be inner regular. The chapter presents a special case of Kolmogorov’s Existence theorem, named for Andrey Kolmogorov.
