Nonstandard measure theory: avoiding pathological sets

Author:
Frank Wattenberg

Journal:
Trans. Amer. Math. Soc. **250** (1979), 357-368

MSC:
Primary 03H05; Secondary 26E35, 28A12, 28A75

DOI:
https://doi.org/10.1090/S0002-9947-1979-0530061-4

MathSciNet review:
530061

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 Inf*mA*|*A* standard,* = 0 and *Q*-thin if Inf**mA*|*A* internal, = 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 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.

**[1]**Allen R. Bernstein and Frank Wattenberg,*Nonstandard measure theory*, Applications of Model Theory to Algebra, Analysis, and Probability, Holt, New York, 1969, pp. 171-185. MR**0247018 (40:287)****[2]**Leo Breiman,*Probability*, Addison-Wesley, Reading, Mass., 1968. MR**0229267 (37:4841)****[2A]**C. Goffman, C. J. Neugebauer and T. Nishiura,*Density topology and approximate continuity*, Duke Math. J.**28**(1961), 497-505. MR**0137805 (25:1254)****[3]**Ward Henson,*On the nonstandard representation of measures*, Trans. Amer. Math. Soc.**172**(1972), 437-446. 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), 113-122. MR**0390154 (52:10980)****[5]**Moshe Machover and Joram Hirschfeld,*Lectures on non-standard analysis*, Lecture Notes in Math., vol. 94, Springer-Verlag, Berlin, 1969. MR**0249285 (40:2531)****[6]**Rohit Parikh and Milton Parnes,*Conditional probability can be defined for all pairs of sets of reals*, Advances in Math.**9**(1972), 313-315. MR**0324736 (48:3085)****[7]**-,*Conditional probabilities and uniform sets*, in Hurd and Loeb, Victoria Symposium on Nonstandard Analysis, Lecture Notes in Math., no. 369, Springer-Verlag, Berlin, 1974, pp. 180-194. MR**0482898 (58:2937)****[8]**Abraham Robinson,*Non-standard analysis*, North-Holland, Amsterdam, 1974. MR**0205854 (34:5680)****[9]**Halsey L. Royden,*Real analysis*, Macmillan, New York, 1968. MR**0151555 (27:1540)****[10]**Stanislaw Saks,*Theory of the integral*, Dover, New York, 1964. MR**0167578 (29:4850)****[11]**Robert Solovay,*A model of set theory in which every set of reals is Lebesgue measurable*, Ann. of Math. (2)**92**(1970), 1-56. 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 theory-Hausdorff measure*, Proc. Amer. Math. Soc.**65**(1977), 326-331. MR**0444466 (56:2817)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
03H05,
26E35,
28A12,
28A75

Retrieve articles in all journals with MSC: 03H05, 26E35, 28A12, 28A75

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0530061-4

Article copyright:
© Copyright 1979
American Mathematical Society