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

Abstract

IfC is a Polish probability space,$W \subseteq \omega ^\omega \times \mathbb{C}$ a Borel set whose sectionsW _x (χ∈ω^ω have measure one and are decreasing$(x \leqslant x\prime \to W_x \supseteq W_{x\prime } )$, then we show that the set ∩ _x W _x has measure one. We give two proofs of this theorem—one in the language of set theory, the other in the language of probability theory, and we apply the theorem to a question on completely uniformly distributed sequences.