# zbMATH — the first resource for mathematics

Remarks on the zero-one law. (English) Zbl 0599.60037
Proofs of the following two known theorems [see J. C. Oxtoby, Measure and category. (1971; Zbl 0217.092)] are outlined. Theorem A: If $$A\subset [0,1)$$ is a Lebesgue measurable ”tail set”, then the Lebesgue measure of A is either 0 or 1. Theorem B: If $$A\subset [0,1)$$ is a ”tail set” possessing the Baire property, then either A or (0,1)$$\setminus A$$ is a set of first Baire category.
The main objective of this paper is to show that the assumptions that A is measurable in Theorem A and that A is a Baire set in Theorem B are not redundant. Two proofs are given, one using standard analysis and the other using nonstandard analysis, of the fact that if $$A\subset [0,1)$$ is a ”tail set”, then A need not be Lebesgue measurable, nor a Baire set.
##### MSC:
 60F20 Zero-one laws 03H10 Other applications of nonstandard models (economics, physics, etc.)
##### Keywords:
Baire property; first Baire category; nonstandard analysis
Full Text:
##### References:
 [1] BILLINGSLEY P.: Probability and measure. Wiley, New York, 1979. · Zbl 0411.60001 [2] DAVIS M.: Applied Nonstandard Analysis. Wiley, New York, 1977. · Zbl 0359.02060 [3] LUXEMBURG W., STROYAN K.: Introduction to the Theory of Infinitesimals. Academic Press, New York, 1976. · Zbl 0336.26002 [4] OXTOBY J.: Measure and Category. Springer-Verlag, New York, 1971. · Zbl 0217.09201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.