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

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.