Advertisement

Borel Spaces and Polish Alphabets

  • Robert M. Gray

Abstract

We have seen that standard measurable spaces are the only measurable spaces for which all finitely additive candidate probability measures are also countably additive, and we have developed several properties and some important simple examples. In particular, sequence spaces drawn from countable alphabets and certain subspaces thereof are standard. In this chapter we develop the most important (and, in a sense, the most general) class of standard spaces-Borel spaces formed from complete separable metric spaces. We will accomplish this by showing that such spaces are isomorphic to a standard subspace of a countable alphabet sequence space and hence are themselves standard. The proof will involve a form of coding or quantization.

Keywords

Limit Point Cauchy Sequence Polish Space Measurable Scheme Countable Union 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. J. Bjornsson, “A note on the characterization of standard borel spaces,” Math. Scand., vol. 47, pp. 135–136, 1980.MathSciNetGoogle Scholar
  2. 2.
    N. Bourbaki, Elements de Mathematique, Livre VI, Integration, HermAnn. Paris, 1956–1965.Google Scholar
  3. 3.
    J. P. R. Christensen, Topology and Borel Structure, Mathematics Studies 10, North-Holland/American Elsevier, New York, 1974.MATHGoogle Scholar
  4. 4.
    D. C. Cohn, Measure Theory, Birkhauser, New York, 1980.MATHGoogle Scholar
  5. 5.
    G. Mackey, “Borel structures in groups and their duals,” Trans. Am. Math. Soc., vol. 85, pp. 134–165, 1957.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967.MATHGoogle Scholar
  7. 7.
    W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1964.MATHGoogle Scholar
  8. 8.
    L. Schwartz, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Oxford University Press, Oxford, 1973.MATHGoogle Scholar
  9. 9.
    G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, New York, 1963.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

  • Robert M. Gray
    • 1
  1. 1.Department of Electrical EngineeringStanford UniversityStanfordUSA

Personalised recommendations