Semipullbacks of labelled Markov processes

A labelled Markov process (LMP) consists of a measurable space S together with an indexed family of Markov kernels from S to itself. This structure has been used to model probabilistic computations in Computer Science, and one of the main problems in the area is to define and decide whether two LMP S and S′ “behave the same”. There are two natural categorical definitions of sameness of behavior: S and S′ are bisimilar if there exist an LMP T and measure preserving maps forming a diagram of the shape S ← T → S′; and they are behaviorally equivalent if there exist some U and maps forming a dual diagram S → U ← S′. These two notions differ for general measurable spaces but Edalat proved that they coincide for analytic Borel spaces, showing that from every diagram S → U ← S′ one can obtain a bisimilarity diagram as above. Moreover, the resulting square of measure preserving maps is commutative (a semipullback). In this paper, we extend Edalat’s result to measurable spaces S isomorphic to a universally measurable subset of a Polish space with the trace of the Borel σ-algebra, using a version of Strassen’s theorem on common extensions of finitely additive measures.