Assume that X is a topological space and Y is a separable metric space. Let these spaces be equipped with Borel σ-algebras B_X and B_Y, respectively. Suppose that P(x, B) is a stochastic transition kernel; i.e., the mapping x ↦ P(x,B) is measurable for all B ∈ B_Y and the mapping B ↦ P(x, B) is a probability measure for any x ∈ X. Denote by supp(P(x, ·) the topological support of the measure B ↦ P(x,B). If the transition kernel P(x,B) satisfies the Feller property, i.e., the mapping x ↦ P(x, ·) is continuous in the weak topology on the space of probability measures, then the set-valued mapping x ↦ supp(P(x, ·) is lower semicontinuous. Conversely, consider a set-valued mapping x ↦ S(x), where x ∈ X and S(x) is a nonempty closed subset of a Polish space Y. If x ↦ S(x) is lower semicontinuous, then, under some general assumptions on the space X, there exists a Feller transition kernel such that supp(P(x, ·) = S(x) for all x ∈ X.

中文翻译：

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集值映射提供度量支持的Feller转换内核

假设X是一个拓扑空间，Y是一个可分离的度量空间。让这些空间配备波雷尔σ -代数乙_X和乙_ÿ，分别。假设P（x，B）是随机转移核；即，映射x↦P（x，B）对于所有B∈B _Y都是可测量的，而映射B↦P（x，B）是任何x∈X的概率度量。用supp（P（x，· ）度量B↦P的拓扑支持（x，B）。如果过渡内核P（X，B）满足费勒属性，即，映射X↦P （X，·）是在对概率测度的空间，则该组值映射弱拓扑连续X ↦SUPP（P（X，·）是下半，相反，考虑一组值映射X↦小号（X），其中的x∈X和小号（X）是一个波兰空间的非空子闭子集ý。如果X↦小号（X）是下半连续的，则在空间X的一些一般假设下，存在一个Feller过渡核，使得对所有x∈X的supp（P（x，·）= S（x）。