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Concept# Polish space

Summary

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.
Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and the Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e.g., the open interval is Polish.
Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that

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A standard question arising in optimal transport theory is whether the Monge problem and the Kantorovich relaxation have the same infimum; the positive answer means that we can pass to the relaxed problem without loss of information. In the classical case with two marginals, this happens when the cost is positive, continuous, and possibly infinite and the first marginal has no atoms. We study a similar multimarginal symmetric problem, arising naturally in density functional theory, motivated by a recent paper by Buttazzo, De Pascale, and Gori Giorgi. The cost is the potential interaction between n charged particles (hence, it is symmetric, positive, continuous, and infinite whenever x(i) = x(j)), and the marginals are all equal with no atoms. We prove that also in this case, there is equality between the infimum in the cyclical Monge problem (the natural Monge problem in this context) and in the classical Kantorovich problem. This result is new even for 2 marginals, because we consider only transport maps which are involutions. The result is generalized to every symmetric continuous cost function on a Polish space.

2015In this paper we make a survey of some recent developments of the theory of Sobolev spaces W-1,W-q (X, d, m), 1 < q < infinity, in metric measure spaces (X, d, m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Gamma-convergence; this result extends Cheeger's work because no Poincare inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of m. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.

2015