Kolmogorov Extension, Martingale Convergence, and Compositionality of Processes

Abstract

We show that the Kolmogorov extension theorem and the Doob martingale convergence theorem are two aspects of a common generalization, namely a colimit-like construction in a category of Radon spaces and reversible Markov kernels. The construction provides a compositional denotational semantics for lossless iteration in probabilistic programming languages, even in the absence of a natural partial order.