Categorical probability spaces, ergodic decompositions, and transitions to equilibrium

Understand

We study a category of probability spaces and measure-preserving Markov kernels up to almost sure equality.

• This category contains, among its isomorphisms, mod-zero isomorphisms of probability spaces.
• It also gives an isomorphism between the space of values of a random variable and the sigma-algebra that it generates on the outcome space, reflecting the standard mathematical practice of using the two interchangeably, for example when taking conditional expectations.
• We show that a number of constructions and results from classical probability theory, mostly involving notions of equilibrium, can be expressed and proven in terms of this category.