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 int...

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Published inInformation and computation Vol. 256; pp. 108 - 130
Main Authors Gogacz, Tomasz, Michalewski, Henryk, Mio, Matteo, Skrzypczak, Michał
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.10.2017
Elsevier
Subjects
Computer Science
Logic in Computer Science
Measure theory
Regular tree languages
Topological complexity
Regular tree languages
Measure theory
Topological complexity
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Summary: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 ω1 ranks. We also expose a connection between regular sets and the σ-algebra of R-sets, introduced by A. Kolmogorov in 1928 as a foundation for measure theory. We show that the game tree languagesWi,k are Wadge-complete for the finite levels of the hierarchy of R-sets. We apply these results to answer positively an open problem regarding the game interpretation of the probabilistic μ-calculus.
ISSN:0890-5401
1090-2651
DOI:10.1016/j.ic.2017.04.012