Geodesic
On Convex Hulls of Compact Sets of Probability Measures with Countable Supports
Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 83-88.

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E. Michael and I. Namioka proved the following theorem. Let $Y$ be a convex $G_\delta$-subset of a Banach space $E$ such that if $K\subset Y$ is a compact space, then its closed (in $Y$) convex hull is also compact. Then every lower semicontinuous set-valued mapping of a paracompact space $X$ to $Y$ with closed (in $Y$) convex values has a continuous selection. E. Michael asked the question: Is the assumption that $Y$ is $G_\delta$ essential? In this note we give an affirmative answer to this question of Michael.
Keywords: continuous selection, set-valued mapping, lower semicontinuity
Mots-clés : paracompact space. 
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V. L. Gejnts; V. V. Filippov. On Convex Hulls of Compact Sets of Probability Measures with Countable Supports. Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 83-88. http://geodesic.mathdoc.fr/item/FAA_2011_45_1_a7/

[1] E. A. Michael, Ann. of Math., 63 (1956), 361–382 | DOI | MR | Zbl

[2] E. A. Michael, Open Problems in Topology, North Holland, Amsterdam, 1990, 271–278 | MR

[3] V. V. Filippov, Comment. Math. Univ. Carolinae, 45:4 (2004), 735–737 | MR | Zbl

[4] V. V. Filippov, Matem. zametki, 78:4 (2005), 638–640 | DOI | MR | Zbl

[5] I. Namioka, E. A. Michael, Topology Appl., 155:8 (2008), 858–860 | DOI | MR | Zbl

[6] U. Rudin, Funktsionalnyi analiz, Mir, M., 1975 | MR

[7] V. V. Fedorchuk, V. V. Filippov, Obschaya topologiya. Osnovnye konstruktsii, Izd-vo MGU, M., 1988 | Zbl