Probability Measures on Product Spaces

Abstract

In order to model a random time evolution, the canonical procedure is to construct probability measures on product spaces. Roughly speaking, the first step is to take a probability measure that models the initial distribution. In the second step, on a different probability space, the distribution after one time step is modeled. Then in each subsequent step, on a further probability space, the random state in the next time step given the full history is modeled. On a formal level, we consider products of probability spaces and Markov kernels between such spaces. Finally, the Ionescu-Tulcea theorem shows that the whole procedure can be realized on a single infinite product space. Furthermore, Kolmogorov’s extension theorem shows that a similar construction can be performed even if the time set is not discrete.