Egoroff's theorem and the distribution of standard points in a nonstandard model
Authors:
C. Ward Henson and Frank Wattenberg
Journal:
Proc. Amer. Math. Soc. 81 (1981), 455461
MSC:
Primary 03H05; Secondary 26E35, 28A12
MathSciNet review:
597662
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Abstract: We study the relationship between the Loeb measure of a set and the measure of the set of standard points in . If is in the algebra generated by the standard sets, then . This is used to give a short nonstandard proof of Egoroff's Theorem. If is an internal, * measurable set, then in general there is no relationship between the measures of and . However, if is an ultrapower constructed using a minimal ultrafilter on , then implies that is a null set. If, in addition, is a Borel measure on a compact metric space and is a Loeb measurable set, then where and are the inner and outer measures for .
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 D. Booth, Ultrafilters on a countable set, Ann. Math. Logic 2 (1970), 124. MR 0277371 (43:3104)
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 P. 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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 A. R. D. Mathias, Solution of problems of Chaquet and Puritz, Conference in Mathematical Logic , Lecture Notes in Math., vol. 255, SpringerVerlag, Berlin and New York, 1972, pp. 204210. MR 0363911 (51:166)
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 K. R. Milliken, Completely separable families and Ramsey's theorem, J. Combinatorial Theory 19 (1975), 318334. MR 0403983 (53:7792)
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 A. Robinson, Nonstandard analysis, NorthHolland, Amsterdam, 1974. MR 0205854 (34:5680)
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 M. Rudin, Types of ultrafilters, Topology Seminar, Wisconsin, R. H. Bing and R. J. Bean (eds.), Princeton Univ. Press, Princeton, N.J., 1966. MR 0216451 (35:7284)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198105976623
PII:
S 00029939(1981)05976623
Article copyright:
© Copyright 1981
American Mathematical Society
