2018/10/03 by Matthijs Vákár, Vákár, Matthijs, C.-H. Luke Ong +1 · 2 citations
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Mathematical Analysis and Transform Methods #Probability (math.PR) #Stochastic processes and financial applications #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1810.01837
openalex publication_date 2018/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note, we develop some of the basic theory of s-finite (measures and) kernels, a little-studied class that Staton has recently argued convincingly to be precisely the semantic counterpart of (first-order) probabilistic programs. We discuss their Carathéodory extension and extend Staton's analysis of their product measures. We give various characterisations of such kernels and discuss their relationship to the more commonly studied classes of σ-finite, subprobability and probability kernels. We use these characterisations to establish suitable Radon-Nikodým, Lebesgue decomposition and disintegration theorems for s-finite kernels. We discuss s-finite analogues of the classical randomisation lemma for probability kernels. Throughout, we give some examples to explain the connection with (first-order) probabilistic programming. Finally, we briefly explore how some of these results extend to quasi-Borel spaces, and hence how they apply to higher-order probabilistic programming.