2023/10/06 by Noé Ensarguet, Paolo Perrone, Ensarguet, Noé +1
Economics, Econometrics and Finance · Mathematics · #18M05 #18M40 #37A50 #60A05 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Category Theory (math.CT) #Dynamical Systems (math.DS) #Economic theories and models #FOS: Computer and information sciences #FOS: Mathematics #Logic in Computer Science (cs.LO) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2310.04267
openalex publication_date 2023/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.