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

IfC is a Polish probability space, a Borel set whose sectionsW_x ( have measure one and are decreasing, then we show that the set _xW_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.Supported by DFG grant Ko 490/7-1.