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Uniqueness of representing measures.

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1757)

Abstract

The question of uniqueness of representing measures is a natural one, both in applications and in the theory itself. As always, one must specify clearly the context within which uniqueness is being asserted. What we would like most is a theorem which characterizes those compact convex X with the property that to each point there exists a unique measure that represents the point and is supported by the extreme points of X. Choquet has proved such a theorem for metrizable X, but there is no satisfactory result in the general case. On the other hand, Choquet and Meyer have characterized those X with the property that to each point there corresponds a unique maximal measure which represents the point. Since maximal measures are “supported” by the extreme points, it would seem that this answers the question, but the fact that “supported” is taken in an approximate sense makes a considerable difference. An example by Mokobodzki will show that uniqueness of maximal representing measures does not imply uniqueness of representing measures which vanish on Baire subsets of X ex X.

Keywords

  • Extreme Point
  • Unique Measure
  • Maximal Measure
  • Compact Hausdorff Space
  • Nonnegative Measure

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). Uniqueness of representing measures.. In: Phelps, R.R. (eds) Lectures on Choquet’s Theorem. Lecture Notes in Mathematics, vol 1757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48719-0_10

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  • DOI: https://doi.org/10.1007/3-540-48719-0_10

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41834-4

  • Online ISBN: 978-3-540-48719-7

  • eBook Packages: Springer Book Archive