Sometimes they can be seen as conditions for which a probability measure, usually an iid one, is ergodic (that is, it is zero-one on the invariant sets of some action).

Statements

Hewitt-Savage zero-one law

Let $(X,p)$ be a probability space. Consider the infinite product $X^\mathbb{N}$, and form the iid measure$\tilde{p}$ on $X^\mathbb{N}$ whose marginals are given by $p$.

(The equivalent characterizations of zero-one measures give equivalent formulations of this statement.)

Kolmogorov’s zero-one law

Let $(X,p)$ be a probability space. Consider the infinite product $X^\mathbb{N}$, and form the iid measure$\tilde{p}$ on $X^\mathbb{N}$ whose marginals are given by $p$.

Then for every tail event$A\subseteq X^\mathbb{N}$ we have that either $\tilde{p}(A)=0$ or $\tilde{p}(A)=1$. That is, $\tilde{p}$ restricted to the tail sigma-algebra is a zero-one measure.

(The equivalent characterizations of zero-one measures give equivalent formulations of this statement.)

Tobias Fritz and Eigil Fjeldgren Rischel, Infinite products and zero-one laws in categorical probability, Compositionality 2(3) 2020. (arXiv:1912.02769)

Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267