An application of Shoenfield's absoluteness theorem to the theory of uniform distribution

Details An application of Shoenfield's absoluteness theorem to the theory of uniform distribution An application of Shoenfield's absoluteness theorem to the theory of uniform distribution

1993-08-06

arxiv.org/abs/math/9308201v1

Mathematics

Logic

Scientific paper

If (B_x: x in N) is a Borel family of sets, indexed by the Baire space N = omega^omega, all B_x have measure zero, and the family is increasing, then the union of all B_x also has measure zero. We give two proofs of this theorem: one in the language of set theory, using Shoenfield's theorem on Sigma-1-2 sets, the other in the language of probability theory, using von Neumann's selection theorem, and we apply the theorem to a question on completely uniformly distributed sequences.

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