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On Prohorov’s theorem for transition probabilities. (English) Zbl 0732.60007

Let S be a Suslin space, \({\mathcal B}(S)\) be the family of Borel subsets of S and (T,\({\mathcal T},\mu)\) be a complete \(\sigma\)-finite measure space. \({\mathcal R}(T,S)\) denotes the set of all transition probabilities from (T,\({\mathcal T})\) into (S,\({\mathcal B}(S))\) endowed with the topology induced by all functionals \(I_ g: {\mathcal R}(T,S)\to R\) given by \[ I_ g(\delta)=\int_{T}[\int_{S}g(t,s)\delta (t)(ds)]\mu (dt), \] where g runs through all \({\mathcal T}\times {\mathcal B}(S)\) measurable real-valued functions, such that g(t,\(\cdot)\) is continuous on S for every \(t\in T\) and \(\sup_{s\in S}| g(t,S)| \leq \Phi_ g(t)\) for all \(t\in T\) and a \(\mu\)-integrable \(\Phi_ g\). If T is a singleton and \(\mu (T)>0\), then \({\mathcal R}(T,S)\) with the above topology (called the weak topology) and the space \({\mathcal P}(S)\) of all Radon probabilities on (S,\({\mathcal B}(S))\) with the narrow topology can be identified. A subset \({\mathcal R}_ 0\) of \({\mathcal R}(T,S)\) is said to be tight if there is a function h: \(T\times S\to [0,\infty]\) such that for every t the set \(\{s\in S:\;h(t,s)\leq \beta \}\) is compact and \(\sup_{\delta \in {\mathcal R}_ 0}I_ h(\delta)<\infty\) (\(\beta \in R\) is arbitrary). The following result - that is a generalization of the classical Prohorov theorem - is proved: If \({\mathcal R}_ 0\subset {\mathcal R}(T,S)\) is tight, then it is relatively weakly compact and relatively weakly sequentially compact. - The proof is mainly based on a reduction of the general problem to the case of metrizable and Suslin S. Then, a previous result of the author [SIAM J. Control Optimization 22, 570-598 (1984; Zbl 0549.49005)] is applied.

MSC:

60B05 Probability measures on topological spaces
46E27 Spaces of measures
60J35 Transition functions, generators and resolvents
28A33 Spaces of measures, convergence of measures
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)

Citations:

Zbl 0549.49005
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