A noncompact Choquet theorem

Edgar, G. A.

doi:10.1090/S0002-9939-1975-0372586-2

A noncompact Choquet theorem
HTML articles powered by AMS MathViewer

by G. A. Edgar

Proc. Amer. Math. Soc. 49 (1975), 354-358

DOI: https://doi.org/10.1090/S0002-9939-1975-0372586-2

PDF | Request permission

Abstract:

The following noncompact analog of Choquet’s theorem is proved. Let $E$ be a Banach space with the Radon-Nikodým property, let $C$ be a separable, closed, bounded, convex subset of $E$, and let a be a point in $C$. Then there is a probability measure $\mu$ on the universally measurable sets in $C$ such that $a$ is the barycenter of $\mu$ and the set of extreme points of $C$ has $\mu$-measure 1.

References

C. Bessaga and A. Pełczyński, On extreme points in separable conjugate spaces, Israel J. Math. 4 (1966), 262–264. MR 211244, DOI 10.1007/BF02771641
Richard D. Bourgin, Barycenters of measures on certain noncompact convex sets, Trans. Amer. Math. Soc. 154 (1971), 323–340. MR 271701, DOI 10.1090/S0002-9947-1971-0271701-X
S. D. Chatterji, Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scand. 22 (1968), 21–41. MR 246341, DOI 10.7146/math.scand.a-10868
Einar Hille, Methods in classical and functional analysis, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0463863
L. H. Loomis, Dilations and extremal measures, Advances in Math. 17 (1975), no. 1, 1–13. MR 374866, DOI 10.1016/0001-8708(75)90081-X
Arthur Lubin, Extensions of measures and the von Neumann selection theorem, Proc. Amer. Math. Soc. 43 (1974), 118–122. MR 330393, DOI 10.1090/S0002-9939-1974-0330393-X
Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288
John von Neumann, On rings of operators. Reduction theory, Ann. of Math. (2) 50 (1949), 401–485. MR 29101, DOI 10.2307/1969463
Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
R. R. Phelps, Dentability and extreme points in Banach spaces, J. Functional Analysis 17 (1974), 78–90. MR 0352941, DOI 10.1016/0022-1236(74)90005-6