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In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.
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About the metrizability of the space of Probability measures $\mathcal{P}(S)$
I'm not really sure what Villani wrote in his monograph, but it is true that one needs to prove that the weak topology is induced by a distance, as a priori it could be another topology with the same …