Infinite products and zero-one laws in categorical probability
Tobias Fritz, Eigil Fjeldgren Rischel
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:munder><mml:mo>⨂</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>J</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math> come in two versions: a weaker but more general one for families of objects <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>J</mml:mi></mml:mrow></mml:msub></mml:math> 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.