Lattices of random sets and progressivity
Let \(T\) be a complete separable metric space and let \({\mathcal B}\) be the Borel-\(\sigma\)-algebra on \(T\). Let \({\mathcal A}\) be a lattice of closed subsets of \(T\) which is closed under countable intersections and which satisfies a separability assumption. Let \((\Omega,{\mathcal F})\) be a measurable space. The authors define the notions of (simple) random set, of (simple) stopping set with values in \({\mathcal A}\) and of the \(\sigma\)-algebra of progressively measurable sets in \({\mathcal F} \otimes {\mathcal B}\) (the latter two notions with respect to a family of sub-\(\sigma\)-algebras of \({\mathcal F}\) indexed by the elements of \({\mathcal A}\)). The definition of stopping sets given by the authors generalizes the notion of co-compatible sets given by \textit{Yu. A. Rozanov} [``Markov random fields'' (1982; Zbl 0498.60057)], but is not the same as the definition given by \textit{T. G. Kurtz} [Ann. Probab. 8, 675-681 (1980; Zbl 0442.60045)]. The authors show that a random set is a stopping set iff the associated set in the product space \({\mathcal F} \otimes {\mathcal B}\) is progressively measurable.
sigma-algebra of progressively measurable sets
scientific article; zbMATH DE number 724076
scientific article