Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In measure theory Prokhorov's theorem relates tightness of measures to weak compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilevich Prokhorov, who considered probability measures on complete separable metric spaces. The term “Prokhorov's theorem” is also applied to later generalizations to either the direct or the inverse statements. Let (S, ) be a separable metric space. Let mathcal{P}(S) denote the collection of all probability measures defined on S (with its Borel -algebra). Since Prokhorov's theorem expresses tightness in terms of compactness, the Arzela-Ascoli theorem is often used to substitute for compactness: in function spaces, this leads to a characterization of tightness in terms of the modulus of continuity or an appropriate analogue — see tightness in classical Wiener space and tightness in Skorokhod space.

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