Nonstandard measure theory: avoiding pathological sets

Wattenberg, Frank

doi:10.1090/S0002-9947-1979-0530061-4

Nonstandard measure theory: avoiding pathological sets
HTML articles powered by AMS MathViewer

by Frank Wattenberg PDF

Trans. Amer. Math. Soc. 250 (1979), 357-368 Request permission

Abstract:

The main results in this paper concern representing Lebesgue measure by nonstandard measures which avoid certain pathological sets. An (external) set E is S-thin if InfmA|A standard,* $A \supseteq E$ = 0 and Q-thin if Inf*mA|A internal, $A \supseteq E$ = 0. It is shown that any *finite sample which represents Lebesgue measure avoids every S-thin set and that given any Q-thin set E there is a *finite sample avoiding E which represents Lebesgue measure. In the last part of the paper a particular pathological set ${\mathcal {H}} \subseteq * \left [ {0, 1} \right ]$ 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 $\sigma$-algebra including ${\mathcal {H}}$ in such a way that ${\mathcal {H}}$ is assigned zero measure.

References

Allen R. Bernstein and Frank Wattenberg, Nonstandard measure theory, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 171–185. MR 0247018
Leo Breiman, Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
Casper Goffman, C. J. Neugebauer, and T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497–505. MR 137805
C. Ward Henson, On the nonstandard representation of measures, Trans. Amer. Math. Soc. 172 (1972), 437–446. MR 315082, DOI 10.1090/S0002-9947-1972-0315082-2
Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
Moshé Machover and Joram Hirschfeld, Lectures on non-standard analysis, Lecture Notes in Mathematics, Vol. 94, Springer-Verlag, Berlin-New York, 1969. MR 0249285, DOI 10.1007/BFb0101447
R. Parikh and M. Parnes, Conditional probability can be defined for all pairs of sets of reals, Advances in Math. 9 (1972), 313–315. MR 324736, DOI 10.1016/0001-8708(72)90022-9
Rohit Parikh and Milton Parnes, Conditional probabilities and uniform sets, Victoria Symposium on Nonstandard Analysis (Univ. Victoria, Victoria, B.C., 1972) Lecture Notes in Math., Vol. 369, Springer, Berlin, 1974, pp. 180–194. MR 0482898
Abraham Robinson, Non-standard analysis, North-Holland Publishing Co., Amsterdam, 1966. MR 0205854
H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555
Stanisław Saks, Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. English translation by L. C. Young; With two additional notes by Stefan Banach. MR 0167578
Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1–56. MR 265151, DOI 10.2307/1970696
K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Pure and Applied Mathematics, No. 72, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0491163
Frank Wattenberg, Nonstandard measure theory–Hausdorff measure, Proc. Amer. Math. Soc. 65 (1977), no. 2, 326–331. MR 444466, DOI 10.1090/S0002-9939-1977-0444466-7