Balder, E. J. An extension of Prohorov’s theorem for transition probabilities with applications to infinite-dimensional lower closure problems. (English) Zbl 0606.60006 Rend. Circ. Mat. Palermo, II. Ser. 34, 427-447 (1985). The paper deals with results concerning the relative weak compactness property of sets of transition probabilities on \(T\times B(S)\), T being a space equipped with a fixed measure and B(S) denoting the \(\sigma\)- algebra of a standard Borel space S. The lower closure and lower semicontinuity theorems play a prominent role in the existence theory for optimal control and have been studied by many authors. The given result generalizes the previous one of the author [SIAM J. Control Optimization 22, 570-598 (1984; Zbl 0549.49005)] and can particularly be of use in obtaining existence theorems for the optimal control of distributed parameter systems. Reviewer: O.Nikonov Cited in 15 Documents MSC: 60B05 Probability measures on topological spaces 49J15 Existence theories for optimal control problems involving ordinary differential equations 49J20 Existence theories for optimal control problems involving partial differential equations 49J45 Methods involving semicontinuity and convergence; relaxation 28A33 Spaces of measures, convergence of measures Keywords:compactness property of sets of transition probabilities; semicontinuity theorems; existence theory for optimal control Citations:Zbl 0549.49005 PDFBibTeX XMLCite \textit{E. J. Balder}, Rend. Circ. Mat. Palermo (2) 34, 427--447 (1985; Zbl 0606.60006) Full Text: DOI References: [1] R. J. Aumann and M. Perles,A variational problem arising in economics, J. Math Anal, Appl.11 (1965), 488–503. · Zbl 0137.39201 [2] Balder E. J.,Lower semicontinuity of integral functionals with nonconvex integrands by relaxation-compactification, SIAM J. 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