🤖 AI Summary
Classical probabilistic constructions—such as almost-invariant σ-algebras, ergodic decompositions, the de Finetti theorem, and the zero–one law—lack a unified structural explanation within standard probability theory.
Method: We construct a category whose objects are probability spaces and whose morphisms are measure-preserving Markov kernels, identified up to almost-sure equality. Our approach integrates categorical methods (notably the Markov category and its dagger structure), standard Borel space theory, and null-isomorphism techniques.
Contribution/Results: We provide the first structural, categorical proof of the ergodic decomposition theorem; characterize almost-invariant σ-algebras uniformly as both limits and colimits; establish a dual limit–colimit characterization of invariant structures in random dynamical systems; and unify three foundational limit theorems—the de Finetti theorem, the zero–one law, and the ergodic decomposition—within a single abstract categorical framework. This advances the structural coherence, universality, and intrinsic categorical nature of probability theory.
📝 Abstract
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. In particular: - Given a stochastic dynamical system acting on a standard Borel space, we show that the almost surely invariant sigma-algebra can be obtained as a limit and as a colimit; - In the setting above, the almost surely invariant sigma-algebra gives rise, up to isomorphism of our category, to a standard Borel space; - As a corollary, we give a categorical version of the ergodic decomposition theorem for stochastic actions; - As an example, we show how de Finetti's theorem and the Hewitt-Savage and Kolmogorov zero-one laws fit in this limit-colimit picture. This work uses the tools of categorical probability, in particular Markov categories, as well as the theory of dagger categories.