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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.)
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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
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