Abstract
The Krein–Milman theorem asserts that in a Hausdorff locally convex space all points of a compact convex set can be approximated by convex combinations of its ‘corners’. We show that this can be reinforced to the statement that all points of the set are barycentres of probability measures living on the closure of the extreme points of the set. An interesting application to completely monotone functions on [0, ∞) yields Bernstein’s theorem concerning Laplace transforms of finite Borel measures on [0, ∞).
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N. Bourbaki: Topologie Générale, Chap. 5 à 10. Hermann, Paris, 1974.
T. Eisner, B. Farkas, M. Haase, and R. Nagel: Operator Theoretic Aspects of Ergodic Theory. Springer, Cham, Switzerland, 2015.
N. J. Kalton and N. T. Peck: A re-examination of the Roberts example of a compact convex set without extreme points. Math. Ann. 253, 89–101 (1980).
M. Krein and D. Milman: On extreme points of regular convex sets. Studia Math. 9, 133–138 (1940).
M. Krein and V. Šmulian: On regularly convex sets in the space conjugate to a Banach space. Ann. Math. 41, 556–583 (1940).
J. Lukeš, J. Malý, I. Netuka, and J. Spurný: Integral Representation Theory. Applications to Convexity, Banach Spaces and Potential Theory. W. de Gruyter, Berlin, 2010.
R. R. Phelps: Lectures on Choquet’s Theorem. 2nd edition. Springer, Berlin, 2001.
J. W. Roberts: Pathological compact convex sets in the spaces L p, 0 < p < 1. The Altgeld Book, Functional Analysis Seminar. University of Illinois, 1976.
J. W. Roberts: A compact convex set with no extreme points. Studia Math. 60, 255–266 (1977).
W. Rudin: Functional Analysis. 2nd edition. McGraw-Hill, New York, 1991.
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Voigt, J. (2020). The Krein–Milman Theorem. In: A Course on Topological Vector Spaces. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-32945-7_17
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DOI: https://doi.org/10.1007/978-3-030-32945-7_17
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-32944-0
Online ISBN: 978-3-030-32945-7
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