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Topology and Measure Theory

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Nonstandard Analysis for the Working Mathematician
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Abstract

We begin this chapter by showing that nonstandard analysis simplifies many of the ideas in the study of metric and topological spaces. After some introductory material, we will present a few more 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.

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References

  1. E.S. Andersen, B. Jessen, Some limit theorems on set-functions. Danske Vid. Selsk. Mat.-Fys. Medd. 25(5), 1–8 (1948)

    Google Scholar 

  2. R.M. Anderson, A nonstandard representation of Brownian motion and Itô integration. Isr. J. Math. 25, 15–46 (1976)

    Article  MATH  Google Scholar 

  3. R.M. Anderson, S. Rashid, A nonstandard characterization of weak convergence. Proc. Am. Math. Soc. 69, 327–332 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. L. Arkeryd, Loeb solutions of the Boltzmann equation. Arch. Ration. Mech. Anal. 86, 85–97 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  5. P.T. Bateman, P. Erdős, Geometrical extrema suggested by a lemma of Besicovitch. Am. Math. Mon. 58, 306–314 (1951)

    Article  MATH  Google Scholar 

  6. V. Bergelson, T. Tao, Multiple recurrence in quasirandom groups. Geom. Funct. Anal. 24, 1–48 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Bliedtner, P.A. Loeb, A reduction technique for limit theorems in analysis and probability theory. Arkiv för Matematik 30(1), 25–43 (1992)

    Google Scholar 

  8. J. Bliedtner, P.A. Loeb, The optimal differentiation basis and liftings of \(L^{{\infty }}\). Trans. Am. Math. Soc. 352, 4693–4710 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Constantinescu, A. Cornea, Ideale Ränder Riemannscher Flächen (Springer, Berlin, 1963)

    Book  MATH  Google Scholar 

  10. N.J. Cutland, S.-A. Ng, The Wiener sphere and Wiener measure. Ann. Probab. 21(1), 1–13 (1993)

    Google Scholar 

  11. J.L. Doob, Stochastic Processes (Wiley, New York, 1953)

    MATH  Google Scholar 

  12. B. Eifrig, Ein Nicht-Standard-Beweis für die Existenz eines Liftings, in Measure Theory Oberwolfach 1975, ed. by A. Bellow. Lecture Notes in Mathematics, vol. 541 (Springer, Berlin, 1976), pp. 133–135

    Google Scholar 

  13. S. Fajardo, H.J. Keisler, Existence theorems in probability theory. Adv. Math. 118, 134–175 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. B. Fuglede, Remarks on fine continuity and the base operation in potential theory. Math. Ann. 210, 207–212 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  15. Z. Fűredi, P.A. Loeb, On the best constant for the Besicovitch covering theorem. Proc. Am. Math. Soc. 121(4), 1063–1073 (1994)

    Article  Google Scholar 

  16. C.W. Henson, The nonstandard hulls of a uniform space. Pac. J. Math. 43, 115–137 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  17. C.W. Henson, Unbounded Loeb measures. Proc. Am. Math. Soc. 74, 143–150 (1979)

    Google Scholar 

  18. C.W. Henson, L.C. Moore, Nonstandard analysis and the theory of Banach spaces, in Nonstandard Analysis: Recent Developments. Springer Lecture Notes in Mathematics, vol. 383 (Springer, Berlin, 1983)

    Google Scholar 

  19. M. Insall, P.A. Loeb, M. Marciniak, End compactifications and general compactifications. J. Log. Anal. 6(7), 1–16 (2014)

    Article  MathSciNet  Google Scholar 

  20. A. Ionescu-Tulcea, C. Ionescu-Tulcea, Topics in the Theory of Lifting (Springer, Berlin, 1969)

    Book  MATH  Google Scholar 

  21. H.J. Keisler, An infinitesimal approach to stochastic analysis. Mem. Am. Math. Soc. 48, 297 (1984)

    Google Scholar 

  22. H.J. Keisler, Infinitesimals in probability theory, in Nonstandard Analysis and Its Applications, ed. by N.J. Cutland (Cambridge Press, Cambridge, 1988), pp. 106–139

    Chapter  Google Scholar 

  23. J.L. Kelley, General Topology (Van Nostrand, New York, 1955)

    MATH  Google Scholar 

  24. P.A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans. Am. Math. Soc. 211, 113–122 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  25. P.A. Loeb, Applications of nonstandard analysis to ideal boundaries in potential theory. Isr. J. Math. 25, 154–187 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  26. P.A. Loeb, A generalization of the Riesz-Herglotz theorem on representing measures. Proc. Am. Math. Soc. 71(1), 65–68 (1978)

    Google Scholar 

  27. P.A. Loeb, Weak limits of measures and the standard part map. Proc. Am. Math. Soc. 77(1), 128–135 (1979)

    Google Scholar 

  28. P.A. Loeb, A construction of representing measures for elliptic and parabolic differential equations. Math. Ann. 260, 51–56 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  29. J. Lukeš, J. Malý, L. Zajíček, Fine Topological Methods in Real Analysis and Potential Theory. Lecture Notes in Mathematics, vol. 1189 (Springer, Berlin, 1986)

    Google Scholar 

  30. W.A.J. Luxemburg, A general theory of monads, in Applications of Model Theory to Algebra, Analysis, and Probability, ed. by W.A.J. Luxemburg (Holt, Rinehart, and Winston, New York, 1969)

    Google Scholar 

  31. E.A. Perkins, A global intrinsic characterization of Brownian local time. Ann. Probab. 9, 800–817 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  32. E.F. Reifenberg, A problem on circles. Math. Gaz. 32, 290–292 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  33. A. Robinson, Compactification of groups and rings and nonstandard analysis. J. Symb. Log. 34, 576–588 (1969)

    Article  MATH  Google Scholar 

  34. A. Robinson, Non-standard Analysis (North-Holland, Amsterdam, 1966)

    MATH  Google Scholar 

  35. S. Salbany, T. Todorov, Nonstandard analysis in topology: nonstandard and standard compactifications. J. Symb. Log. 65, 1836–1840 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  36. Y.N. Sun, Integration of correspondences on Loeb spaces. Trans. Am. Math. Soc. 349, 129–153 (1997)

    Article  MATH  Google Scholar 

  37. Y.N. Sun, A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN. J. Math. Econ. 29, 419–503 (1998)

    Article  MATH  Google Scholar 

  38. T. Tao, T. Ziegler, The inverse Conjecture for the Gowers norm over finite fields in low characteristic. Ann. Comb. 16, 121–188 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  39. T. Tao, Hilbert’s Fifth Problem and Related Topics. Graduate Studies in Mathematics, vol. 153 (American Mathematical Society, Providence, 2014)

    Google Scholar 

  40. F. Wattenberg, Nonstandard measure theory: avoiding pathological sets. Trans. Am. Math. Soc. 250, 357–368 (1979)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Peter A. Loeb .

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Loeb, P.A. (2015). Topology and Measure Theory. In: Loeb, P., Wolff, M. (eds) Nonstandard Analysis for the Working Mathematician. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7327-0_3

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