Tateishi, Hiroshi The Skorokhod representation theorem for Young measures. (English) Zbl 1429.28002 Trans. Am. Math. Soc. 372, No. 9, 6589-6602 (2019). Let \((\Omega, F , \mu)\) be a complete finite positive measure space and let \(\mathbb{S}\) be a Polish space. A probability measure on \((\Omega \times \mathbb{S}, F \otimes \mathcal{B}(\mathbb{S}))\) whose projection on \(\Omega\) is equal to \(\mu\) is called a Young measure, and the space of such measures is denoted by \(\mathcal{Y}(\Omega, \mathbb{S})\). Let \(\nu_n\) \((n \in \mathbb{N} \cup \{0\})\) be probability measures on \(\mathbb{S}\) such that \(\nu_n \rightarrow \nu_0\) in the weak\(^*\) topology. The Skorokhod representation theorem states that there exists a sequence \(\xi_n\) \((n \in \mathbb{N} \cup \{0\})\) of random variables on some probability space such that the law of \(\xi_n\) coincides with \(\nu_n\) and \(\xi_n \rightarrow \xi_0\) almost surely. The purpose of this paper is to extend the Skorokhod representation theorem to Young measures. It is also proved that if \(\mathbb{S}\) and \(\mathbb{T}\) are Souslin spaces and there exists \(\varphi: \mathbb{S} \longrightarrow \mathbb{T}\) a continuous, open, and surjective map, if the space \(\mathcal{Y}(\Omega, \mathbb{S})\) has the Skorokhod representation property, then the space \(\mathcal{Y}(\Omega, \mathbb{T})\) also has the Skorokhod representation property. Reviewer: Daniele Puglisi (Catania) MSC: 28A33 Spaces of measures, convergence of measures 60B10 Convergence of probability measures Keywords:Young measure; Skorokhod representation; stable topology; open mapping PDFBibTeX XMLCite \textit{H. Tateishi}, Trans. Am. Math. Soc. 372, No. 9, 6589--6602 (2019; Zbl 1429.28002) Full Text: DOI References: [1] B E. Balder, Lectures on Young measures, Cahiers de math\'ematiques de la d\'ecision 9517, CEREMADE, Universit\'e Paris\textendash Dauphine, 1995. [2] Blackwell, David; Dubins, Lester E., An extension of Skorohod’s almost sure representation theorem, Proc. Amer. Math. Soc., 89, 4, 691-692 (1983) · Zbl 0542.60005 [3] Banakh, T. O.; Bogachev, V. I.; Kolesnikov, A. V., Topological spaces with the strong Skorokhod property, Georgian Math. J., 8, 2, 201-220 (2001) · Zbl 1023.60004 [4] Bogachev, V. I.; Kolesnikov, A. V., Open mappings of probability measures and Skorokhod’s representation theorem, Teor. Veroyatnost. i Primenen.. Theory Probab. Appl., 46 46, 1, 20-38 (2002) · Zbl 1023.60002 [5] Castaing, Charles; Raynaud de Fitte, Paul; Valadier, Michel, Young measures on topological spaces: With applications in control theory and probability theory, Mathematics and its Applications 571, xii+320 pp. (2004), Kluwer Academic Publishers, Dordrecht · Zbl 1067.28001 [6] Dudley, R. M., Distances of probability measures and random variables, Ann. Math. Statist., 39, 1563-1572 (1968) · Zbl 0169.20602 [7] Fernique, Xavier, Un mod\`“ele presque s\^ur pour la convergence en loi, C. R. Acad. Sci. Paris S\'”{e}r. I Math., 306, 7, 335-338 (1988) · Zbl 0636.60025 [8] Jakubowski, A., The almost sure Skorokhod representation for subsequences in nonmetric spaces, Teor. Veroyatnost. i Primenen.. Theory Probab. Appl., 42 42, 1, 167-174 (1998) (1997) · Zbl 0923.60001 [9] Michael, E., A selection theorem, Proc. Amer. Math. Soc., 17, 1404-1406 (1966) · Zbl 0178.25902 [10] Pedregal, Pablo, Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications 30, xii+212 pp. (1997), Birkh\"{a}user Verlag, Basel · Zbl 0879.49017 [11] Schief, Andreas, An open mapping theorem for measures, Monatsh. Math., 108, 1, 59-70 (1989) · Zbl 0679.28004 [12] Skorokhod, A. V., Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen., 1, 289-319 (1956) · Zbl 0074.33802 [13] Tateishi, Hiroshi, An open mapping theorem for Young measures, Proc. Amer. Math. Soc., 136, 11, 4027-4032 (2008) · Zbl 1160.60003 [14] Valadier, Michel, Young measures. Methods of nonconvex analysis, Varenna, 1989, Lecture Notes in Math. 1446, 152-188 (1990), Springer, Berlin · Zbl 0738.28004 [15] Y L. C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, C. R. Soc. Sci. Lett. Varsovie, Classe III 30 (1937), 212-234. · Zbl 0019.21901 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.