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 Teor. Veroyatnost. i Primenen., 2001, Volume 46, Issue 1, Pages 3–27 (Mi tvp3944)

Open Mappings of Probability Measures and the Skorokhod Representation Theorem

V. I. Bogachev, A. V. Kolesnikov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove that for the wide class of spaces X and Y (including completely regular Souslin spaces), every open surjective mapping $f\colon X\to Y$ induces the open mapping $\hat f\colon\mu\mapsto\mu\circ f^{-1}$ between the spaces of probability measures ${\mathcal P} (X)$ and ${\mathcal P} (Y)$. We discuss the existence of continuous inverse mappings for $\hat f$ and connections with the Skorokhod representation theorem and its generalizations.

Keywords: weak convergence of probability measures, Skorokhod representation, open mapping, continuous selection.

DOI: https://doi.org/10.4213/tvp3944

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English version:
Theory of Probability and its Applications, 2002, 46:1, 20–38

Bibliographic databases:

Citation: V. I. Bogachev, A. V. Kolesnikov, “Open Mappings of Probability Measures and the Skorokhod Representation Theorem”, Teor. Veroyatnost. i Primenen., 46:1 (2001), 3–27; Theory Probab. Appl., 46:1 (2002), 20–38

Citation in format AMSBIB
\Bibitem{BogKol01} \by V.~I.~Bogachev, A.~V.~Kolesnikov \paper Open Mappings of Probability Measures and the Skorokhod Representation Theorem \jour Teor. Veroyatnost. i Primenen. \yr 2001 \vol 46 \issue 1 \pages 3--27 \mathnet{http://mi.mathnet.ru/tvp3944} \crossref{https://doi.org/10.4213/tvp3944} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1968703} \zmath{https://zbmath.org/?q=an:1023.60002} \transl \jour Theory Probab. Appl. \yr 2002 \vol 46 \issue 1 \pages 20--38 \crossref{https://doi.org/10.1137/S0040585X97978701} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000174464700002} 

• http://mi.mathnet.ru/eng/tvp3944
• https://doi.org/10.4213/tvp3944
• http://mi.mathnet.ru/eng/tvp/v46/i1/p3

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This publication is cited in the following articles:
1. Banakh T.O., Bogachev V.I., Kolesnikov A.V., “Topological spaces with Prokhorov and Skorokhod properties”, Dokl. Math., 64:2 (2001), 244–247
2. Banakh T., Chigogidze A., Fedorchuk V., “On spaces of $\sigma$-additive probability measures”, Topology Appl., 133:2 (2003), 139–155
3. Kolesnikov A.V., “Convexity inequalities and optimal transport of infinite-dimensional measures”, J. Math. Pures Appl. (9), 83:11 (2004), 1373–1404
4. V. I. Bogachev, A. V. Kolesnikov, “Integrability of absolutely continuous measure transformations and applications to optimal transportation”, Theory Probab. Appl., 50:3 (2006), 367–385
5. Valov V., “Probability measures and Milyutin maps between metric spaces”, J. Math. Anal. Appl., 350:2 (2009), 723–730
6. Roininen L., Piiroinen P., Lehtinen M., “Constructing Continuous Stationary Covariances as Limits of the Second-Order Stochastic Difference Equations”, Inverse Probl. Imaging, 7:2 (2013), 611–647
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