Nonstandard measure theory: avoiding pathological sets
Author:
Frank Wattenberg
Journal:
Trans. Amer. Math. Soc. 250 (1979), 357368
MSC:
Primary 03H05; Secondary 26E35, 28A12, 28A75
MathSciNet review:
530061
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Abstract: The main results in this paper concern representing Lebesgue measure by nonstandard measures which avoid certain pathological sets. An (external) set E is Sthin if InfmAA standard,* = 0 and Qthin if Inf*mAA internal, = 0. It is shown that any *finite sample which represents Lebesgue measure avoids every Sthin set and that given any Qthin set E there is a *finite sample avoiding E which represents Lebesgue measure. In the last part of the paper a particular pathological set is constructed which is important for the study of approximate limits, derivatives etc. It is shown that every *finite sample which represents Lebesgue measure assigns inner measure zero and outer measure one to this set and that Loeb measure does the same. Finally, it is shown that Loeb measure can be extended to a algebra including in such a way that is assigned zero measure.
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 Leo Breiman, Probability, AddisonWesley, Reading, Mass., 1968. MR 0229267 (37:4841)
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 C. Goffman, C. J. Neugebauer and T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497505. MR 0137805 (25:1254)
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 Ward Henson, On the nonstandard representation of measures, Trans. Amer. Math. Soc. 172 (1972), 437446. MR 0315082 (47:3631)
 [4]
 Peter Loeb, Conversion from nonstandard to standard measure spaces and applications to probability theory, Trans. Amer. Math Soc. 211 (1975), 113122. MR 0390154 (52:10980)
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 Moshe Machover and Joram Hirschfeld, Lectures on nonstandard analysis, Lecture Notes in Math., vol. 94, SpringerVerlag, Berlin, 1969. MR 0249285 (40:2531)
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 Rohit Parikh and Milton Parnes, Conditional probability can be defined for all pairs of sets of reals, Advances in Math. 9 (1972), 313315. MR 0324736 (48:3085)
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 , Conditional probabilities and uniform sets, in Hurd and Loeb, Victoria Symposium on Nonstandard Analysis, Lecture Notes in Math., no. 369, SpringerVerlag, Berlin, 1974, pp. 180194. MR 0482898 (58:2937)
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 Abraham Robinson, Nonstandard analysis, NorthHolland, Amsterdam, 1974. MR 0205854 (34:5680)
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 Halsey L. Royden, Real analysis, Macmillan, New York, 1968. MR 0151555 (27:1540)
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 Stanislaw Saks, Theory of the integral, Dover, New York, 1964. MR 0167578 (29:4850)
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 Robert Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 156. MR 0265151 (42:64)
 [12]
 Keith Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Academic Press, New York, 1976. MR 0491163 (58:10429)
 [13]
 Frank Wattenberg, Nonstandard measure theoryHausdorff measure, Proc. Amer. Math. Soc. 65 (1977), 326331. MR 0444466 (56:2817)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905300614
PII:
S 00029947(1979)05300614
Article copyright:
© Copyright 1979
American Mathematical Society
