Topology and Measure Theory

  • Peter A. Loeb
Part of the Mathematics and Its Applications book series (MAIA, volume 510)


We begin this chapter by showing that nonstandard analysis simplifies many of the ideas in the study of metric and topological spaces. Most of the initial results on topology can be found in Robinson’s book [30]. We then give some recent applications of nonstandard analysis to topology. The chapter concludes with a quick introduction to the applications of nonstandard analysis in measure and probability theory. For this purpose, we assume the reader is familiar with the Carathéodory Extension Theorem. A full introduction to this rich theory, starting from first principles, is given in Horst Osswald’s chapters in Part III of this book.


Topological Space Measure Theory Countable Base Nonstandard Analysis Base Generate Function 
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  1. 1.
    E. S. Andersen and B. Jessen, Some limit theorems on set-functions, Danske Vid. Selsk. Mat.-Fys. Medd. 25(1948), #5, 1–8.Google Scholar
  2. 2.
    R. M. Anderson, A nonstandard representation of Brownian motion and Itô integration, Israel J. Math. 25(1976), 15–46.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    R. M. Anderson and S. Rashid, A nonstandard characterization of weak convergence, Proc. Amer. Math. Soc. 69(1978), 327–332.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    L. Arkeryd, Loeb solutions of the Boltzmann equation, Arch. Rational Mech. Anal. 86(1984), 85–97.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    P. T. Bateman and P. Erdős, Geometrical extrema suggested by a lemma of Besicovitch, Amer. Math. Monthly 58(1951), 306–314.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    J. Bliedtner and P. A. Loeb, A reduction technique for limit theorems in analysis and probability theory, Arkiv för Matematik 30(1992), #1, 25–43.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    —, The optimal differentiation basis and liftings of L , to appear in Trans. Amer. Math. Soc. Google Scholar
  8. 8.
    C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen, Springer, Berlin, 1963.zbMATHCrossRefGoogle Scholar
  9. 9.
    N. J. Cutland and Siu-Ah Ng, The Wiener Sphere and Wiener Measure, Ann. Probability 21(1993), No. 1, 1–13.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    J. L. Doob, Stochastic Processes, Wiley, New York, 1953.zbMATHGoogle Scholar
  11. 11.
    B. Eifrig, Ein Nicht-Standard-Beweis für die Existenz eines Liftings, in: Measure Theory Oberwolf ach 1975, A. Bellow (ed.), Springer-Verlag Lecture Notes in Mathematics 541, Berlin, 1976, 133–135.Google Scholar
  12. 12.
    S. Fajardo and H. J. Keisler, Existence theorems in probability theory, Advances in Mathematics 118(1996), 134–175.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    B. Fuglede, Remarks on fine continuity and the base operation in potential theory, Math Ann. 210(1974), 207–212.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Z. Fűredi and P. A. Loeb, On the best constant for the Besicovitch covering theorem, Proc. Amer. Math. Soc. 121(1994), #4, 1063-1073.Google Scholar
  15. 15.
    C. W. Henson, Unbounded Loeb measures, Proc. Amer. Math. Soc. 74(1979).Google Scholar
  16. 16.
    A. Ionescu-Tulcea and C. Ionescu-Tulcea, Topics in the theory of lifting, Springer-Verlag, New York, 1969.zbMATHCrossRefGoogle Scholar
  17. 17.
    H. J. Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48(1984), no 297.Google Scholar
  18. 18.
    H. J. Keisler —, Infinitesimals in probability theory, in: Nonstandard Analysis and its Applications, N. J. Cutland (ed.), Cambridge Press, Cambridge, 1988, 106–139.Google Scholar
  19. 19.
    J. L. Kelley, General Topology, Van Nostrand, New York, 1955.zbMATHGoogle Scholar
  20. 20.
    P. A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211(1975), 113–122.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    —, Applications of nonstandard analysis to ideal boundaries in potential theory, Israel J. Math. 25(1976), 154–187.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    —, A generalization of the Riesz-Herglotz Theorem on representing measures, Proc. Amer. Math. Soc. 71(1978), # 1, 65–68.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    —, Weak limits of measures and the standard part map, Proc. Amer. Math. Soc. 77(1979), # 1, 128–135.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    —, A construction of representing measures for elliptic and parabolic differential equations, Math. Annalen 260(1982), 51–56.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    J. Lukeš, J. Malý, L. Zajíček, Fine Topological Methods in Real Analysis and Potential Theory, Springer-Verlag Lecture Notes in Mathematics 1189, 1986.Google Scholar
  26. 26.
    W. A. J. Luxemburg, A general theory of monads, in: Applications of Model Theory to Algebra, Analysis, and Probability, W. A. J. Luxemburg (ed.), Holt, Rinehart, and Winston, New York, 1969.Google Scholar
  27. 27.
    E. A. Perkins, A global intrinsic characterization of Brownian local time, Ann. Probability 9(1981), 800–817.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    E. F. Reifenberg, A problem on circles, Math. Gaz. 32(1948), 290–292.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    A. Robinson, Compactification of groups and rings and nonstandard analysis, Jour. of Symbolic Logic 34(1969), 576–588.zbMATHCrossRefGoogle Scholar
  30. 30.
    —, Non-standard Analysis, North-Holland, Amsterdam, 1966.zbMATHGoogle Scholar
  31. 31.
    S. Salbany and T. Todorov, Nonstandard Analysis in Topology: Nonstandard and Standard Compactifications, to appear in J. Symbolic Logic. Google Scholar
  32. 32.
    Y. N. Sun, Integration of correspondences on Loeb spaces, Transactions of the American Mathematical Society 349(1997), 129–153.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    —, A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN, Journal of Mathematical Economics 29(1998), 419–503.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    F. Wattenberg, Nonstandard measure theory: avoiding pathological sets, Trans. Amer. Math. Soc. 250 (1979), 357–368.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Peter A. Loeb
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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