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 $S^{\mathrm{\prime }}$ “behave the same”. There are two natural categorical definitions of sameness of behavior: $S$ and $S^{\mathrm{\prime }}$ are bisimilar if there exist an LMP $T$ and measure preserving maps forming a diagram of the shape $ S\leftarrow T \rightarrow{S’}$;andtheyarebehaviorallyequivalentifthereexistsome$ U$andmapsformingadualdiagram$ S\rightarrow U \leftarrow{S’}$. These two notions differ for general measurable spaces but Doberkat (extending a result by Edalat) proved that they coincide for analytic Borel spaces, showing that from every diagram $S\to U\leftarrow S^{\mathrm{\prime }}$ 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 the previous result to measurable spaces $S$ isomorphic to a universally measurable subset of a Polish space with the trace of the Borel $\sigma$-algebra, using a version of Strassen’s theorem on common extensions of finitely additive measures.