Infinite products and zero-one laws in categorical probability

TL;DR

This paper extends the categorical framework of probability to include infinite products and zero-one laws, providing a general foundation applicable across various probabilistic contexts.

Contribution

It develops a categorical approach to infinite products and Kolmogorov extension theorem within Markov categories, generalizing classical probability results.

Findings

Versions of Kolmogorov and Hewitt-Savage zero-one laws in Markov categories

Generalized infinite tensor products in semicartesian symmetric monoidal categories

Applicability to measure-theoretic and other probabilistic frameworks

Abstract

Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products $\bigotimes_{i \in J} X_i$ come in two versions: a weaker but more general one for families of objects $(X_i)_{i \in J}$ in semicartesian symmetric monoidal categories, and a stronger but more specific one for families of objects in Markov categories. As a first application, we state and prove versions of the zero-one laws of Kolmogorov and Hewitt-Savage for Markov categories. This gives general versions of these results which can be instantiated not only in measure-theoretic probability, where they specialize to the standard ones in the setting of standard Borel spaces, but also in other contexts.