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Two examples concerning uniform convergence of measures w.r.t. balls in Banach spaces

  • Flemming Topsøe
  • Richard M. Dudley
  • Jørgen Hoffmann-Jørgensen
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 566)

Abstract

Two examples show that certain uniform convergence properties related to the Glivenko-Cantelli theorem — properties which are known to hold in Euclidean spaces — need not hold in general Banach spaces.

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References

  1. [1]
    Elker,J.: Unpublished "Diplomarbeit" from the Ruhr university, Bochum 1975.Google Scholar
  2. [2]
    Topsøe, F.: On the Glivenko-Cantelli Theorem. Z. Wahrscheinlichkeitsrechnung verw. Geb. 14, 239–250 (1970).CrossRefzbMATHGoogle Scholar
  3. [3]
    Topsøe, F.: Uniformity in weak convergence w.r.t. balls in Banach spaces. Math. Scand. 38, 148–158 (1976).MathSciNetzbMATHGoogle Scholar
  4. [4]
    Vapnik, V.N., and Červonenkis, A.Ya.: On the uniform convergence of relative frequencies of events to their probabilities. Theor. Probability Appl. 16, 264–280 (1971).CrossRefzbMATHGoogle Scholar
  5. [5]
    Varadarajan, V.S.: On the convergence of probability distributions, Sankyã 19, 23–26 (1958).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Flemming Topsøe
    • 1
    • 2
    • 3
  • Richard M. Dudley
    • 1
    • 2
    • 3
  • Jørgen Hoffmann-Jørgensen
    • 1
    • 2
    • 3
  1. 1.The University of CopenhagenDenmark
  2. 2.Massachusetts Institute of TechnologyUSA
  3. 3.The University of AarhusDenmark

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