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A different method for extending the representation theorems

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

Abstract

When we say that a probability measure μ on a compact convex set X “represents” a point x of X, we mean, of course, that μ(f) = f(x) for each continuous affine function f on X. One way of extending the representation theorems would be to show that this latter equality holds for a larger class of functions. For instance, Proposition 10.7 showed that it holds for upper semicontinuous (or lower semicontinuous) affine functions. In this section we will show that it holds for the affine functions of first Baire class, i.e., those affine functions which are the pointwise limit of a sequence of continuous (but not necessarily affine) functions on X.

Keywords

  • Representation Theorem
  • Compact Convex
  • Induction Step
  • Borel Subset
  • Affine Function

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). A different method for extending the representation theorems. 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_14

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  • DOI: https://doi.org/10.1007/3-540-48719-0_14

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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