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.

翻译：我们研究了一类概率空间与几乎必然相等意义下保持测度的马尔可夫核构成的范畴。该范畴的同构中包含概率空间的模零同构，并建立了随机变量取值空间与其在结果空间上生成的σ-代数之间的同构关系，这体现了经典数学实践中（例如在条件期望的构造中）将二者互换使用的惯例。我们证明，经典概率论中许多涉及平衡态概念的构造与定理，均可在此范畴框架下表述与证明。具体而言：- 对于作用在标准Borel空间上的随机动力系统，我们证明几乎必然不变σ-代数可同时作为极限与余极限获得；- 在上述设定下，几乎必然不变σ-代数在该范畴的同构意义下可导出标准Borel空间；- 作为推论，我们给出随机作用的遍历分解定理的范畴版本；- 通过实例，我们展示了德菲内蒂定理、休伊特-萨维奇零一律以及柯尔莫哥洛夫零一律如何纳入此极限-余极限图景。本研究运用了范畴概率工具（尤其是马尔可夫范畴）以及dagger范畴理论。