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On topological properties of the Skorokhod space. (English. Russian original) Zbl 0946.54029

Theory Probab. Appl. 43, No. 4, 640-644 (1998); translation from Teor. Veroyatn. Primen. 43, No. 4, 781-786 (1998).
If \(X\) is a Suslin (analytic) subset of a Polish space \(E\), this property is inherited by the Skorokhod space \(D_1(X) \subset D_1(E)\) endowed with the topology of pointwise convergence [L. Schwartz, “Radon measures on arbitrary spaces and cylindrical measures” (1973; Zbl 0298.28001)]. As the paper shows, this need not be true when \(D_1(X)\) is endowed with the Skorokhod topology. Even universal measurability might fail for \(D_1(X)\) under the hypothesis that the fourth projective class \(CPCA\) of \([0,1]\) contains a set which is not universally measurable. However, if \(X\) is co-analytic (having Suslin complement) the same is true for \(D_1(X)\).

MSC:

54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
60G05 Foundations of stochastic processes
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54B05 Subspaces in general topology
60B05 Probability measures on topological spaces
28C99 Set functions and measures on spaces with additional structure
54C35 Function spaces in general topology

Citations:

Zbl 0298.28001
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