Measure properties of regular sets of trees

We investigate measure theoretic properties of regular sets of infinite trees. As a first result, we prove that every regular set is universally measurable and that every Borel measure on the Polish space of trees is continuous with respect to a natural transfinite stratification of regular sets into ranks. We also expose a connection between regular sets and the σ-algebra of -sets, introduced by A. Kolmogorov in 1928 as a foundation for measure theory. We show that the game tree languages are Wadge-complete for the finite levels of the hierarchy of -sets. We apply these results to answer positively an open problem regarding the game interpretation of the probabilistic μ-calculus.