Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Conversion from nonstandard to standard measure spaces and applications in probability theory


Author: Peter A. Loeb
Journal: Trans. Amer. Math. Soc. 211 (1975), 113-122
MSC: Primary 28A10; Secondary 02H25, 60J99
DOI: https://doi.org/10.1090/S0002-9947-1975-0390154-8
MathSciNet review: 0390154
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (X,\mathcal{A},\nu )$ be an internal measure space in a denumerably comprehensive enlargement. The set X is a standard measure space when equipped with the smallest standard $ \sigma $-algebra $ \mathfrak{M}$ containing the algebra $ \mathcal{A}$, where the extended real-valued measure $ \mu $ on $ \mathfrak{M}$ is generated by the standard part of $ \nu $. If f is $ \mathcal{A}$-measurable, then its standard part $ ^0f$ is $ \mathfrak{M}$-measurable on X. If f and $ \mu $ are finite, then the $ \nu $-integral of f is infinitely close to the $ \mu $-integral of $ ^0f$. Applications include coin tossing and Poisson processes. In particular, there is an elementary proof of the strong Markov property for the stopping time of the jth event and a construction of standard sample functions for Poisson processes.


References [Enhancements On Off] (What's this?)

  • [1] Allen R. Bernstein and Peter A. Loeb, A non-standard integration theory for unbounded functions, Victoria Symposium on Nonstandard Analysis (Univ. Victoria, Victoria, B.C., 1972) Springer, Berlin, 1974, pp. 40–49. Lecture Notes in Math., Vol. 369. MR 0492167
  • [2] William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1957. 2nd ed. MR 0088081
  • [3] Paul G. Hoel, Sidney C. Port, and Charles J. Stone, Introduction to statistical theory, Houghton Mifflin Co., Boston, Mass., 1971. The Houghton Mifflin Series in Statistics. MR 0358878
    Paul G. Hoel, Sidney C. Port, and Charles J. Stone, Introduction to stochastic processes, Houghton Mifflin Co., Boston, Mass., 1972. The Houghton Mifflin Series in Statistics. MR 0358879
    Paul G. Hoel, Sidney C. Port, and Charles J. Stone, Introduction to probability theory, Houghton Mifflin Co., Boston, Mass., 1971. The Houghton Mifflin Series in Statistics. MR 0358880
  • [4] Peter A. Loeb, A non-standard representation of measurable spaces, 𝐿_{∞}, and 𝐿*_{∞}, Contributions to non-standard analysis (Sympos., Oberwolfach, 1970), North-Holland, Amsterdam, 1972, pp. 65–80. Studies in Logic and Found. Math., Vol. 69. MR 0482128
  • [5] Peter A. Loeb, A nonstandard representation of Borel measures and 𝜎-finite measures, Victoria Symposium on Nonstandard Analysis (Univ. Victoria, Victoria, B.C., 1972) Springer, Berlin, 1974, pp. 144–152. Lecture Notes in Math., Vol. 369. MR 0476992
  • [6] W. A. J. Luxemburg, A general theory of monads, Applications of Model Theory to Algebra, Analysis, and Probability (Inte rnat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR 0244931
  • [7] Abraham Robinson, Non-standard analysis, North-Holland Publishing Co., Amsterdam, 1966. MR 0205854
  • [8] H. L. Royden, Real analysis, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963. MR 0151555

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 28A10, 02H25, 60J99

Retrieve articles in all journals with MSC: 28A10, 02H25, 60J99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0390154-8
Keywords: Measure space, probability space, standard integral, coin tossing, Poisson processes, sample function, strong Markov property
Article copyright: © Copyright 1975 American Mathematical Society