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Extensions and Applications of the S-Measure Construction

Published online by Cambridge University Press:  12 March 2014

David A. Ross
David A. Ross*
Affiliation:
Department of Mathematics, University of Hawai'i, 2565 McCarthy Mall, Honolulu, HI 96822, USA, E-mail: ross@math.hawaii.edu, URL: www.math.hawaii.edu/~ross
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Abstract

S-measures are Loeb measures restricted to the sigma algebra generated by standard sets. This paper gives new extensions of the S-measure machinery, with applications to standard measure theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 4 , December 2013 , pp. 1247 - 1256
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1] Bartle, Robert G., The elements of integration, John Wiley & Sons Inc., New York, 1966.Google Scholar
[2] Henson, C. Ward and Wattenberg, Frank, Egoroff's theorem and the distribution of standard points in a nonstandard model, Proceedings of the American Mathematical Society, vol. 81 (1981), no. 3, pp. 455–461.Google Scholar
[3] Kamae, Teturo, A simple proof of the ergodic theorem using nonstandard analysis, Israel Journal of Mathematics, vol. 42 (1982), no. 4, pp. 284–290.CrossRefGoogle Scholar
[4] Kamae, Teturo and Keane, Michael, A simple proof of the ratio ergodic theorem, Osaka Journal of Mathematics, vol. 34 (1997), no. 3, pp. 653–657.Google Scholar
[5] Loeb, Peter A., Nonstandard analysis and topology, Nonstandard analysis (Arkeryd, Leif O., Cutland, Nigel J., and Henson, C. Ward, editors), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 493, Kluwer Academic Publishers, Dordrecht, 1997, pp. 77–89.CrossRefGoogle Scholar
[6] Loeb, Peter A. and Talvila, Erik, Covering theorems and Lebesgue integration, Scientiae Mathematicae Japonicae, vol. 53 (2001), no. 2, pp. 209–221.Google Scholar
[7] Oxtoby, John C., Measure and category, second ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York, 1980.CrossRefGoogle Scholar
[8] Robinson, Abraham, Non-standard analysis, North-Holland, Amsterdam, 1966.Google Scholar
[9] Ross, David A., Lifting theorems in nonstandard measure theory, Proceedings of the American Mathematical Society, vol. 109 (1990), no. 3, pp. 809–822.CrossRefGoogle Scholar
[10] Ross, David A., Loeb measure and probability, Nonstandard analysis (Arkeryd, Leif O., Cutland, Nigel J., and Henson, C. Ward, editors), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 493, Kluwer Academic Publishers, Dordrecht, 1997, pp. 91–120.CrossRefGoogle Scholar
[11] Ross, David A., Nonstandard measure constructions—solutions and problems, Nonstandard methods and applications in mathematics (Cutland, Nigel J., Nasso, Mauro Di, and Ross, David A., editors), Lecture Notes in Logic, vol. 25, The Association for Symbolic Logic, La Jolla, CA, 2006, pp 127–146.Google Scholar
[12] Ross, David A., More on S-measures, The strength of nonstandard analysis (van den Berg, I. and Neves, V., editors), Springer, Wien, 2007, pp. 217–226.CrossRefGoogle Scholar