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 in | Information and computation Vol. 256; pp. 108 - 130 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
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01.10.2017
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Gogacz, Tomasz Skrzypczak, Michał Michalewski, Henryk Mio, Matteo |
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| Keywords | Regular tree languages Measure theory Topological complexity |
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Sb. contributor: fullname: Kolmogorov – start-page: 431 year: 2008 ident: 10.1016/j.ic.2017.04.012_br0160 article-title: From automatic structures to Borel structures contributor: fullname: Hjorth – volume: 18 start-page: 214 year: 1932 ident: 10.1016/j.ic.2017.04.012_br0220 article-title: Memoir on analytical operations and projective sets publication-title: Fundam. Math. doi: 10.4064/fm-18-1-214-279 contributor: fullname: Kantorovich – volume: 141 start-page: 1 year: 1969 ident: 10.1016/j.ic.2017.04.012_br0320 article-title: Decidability of second-order theories and automata on infinite trees publication-title: Trans. Am. Math. Soc. contributor: fullname: Rabin – start-page: 165 year: 2012 ident: 10.1016/j.ic.2017.04.012_br0290 article-title: On topological completeness of regular tree languages contributor: fullname: Michalewski – start-page: 384 year: 2005 ident: 10.1016/j.ic.2017.04.012_br0090 article-title: Transfinite extension of the mu-calculus contributor: fullname: Bradfield – volume: 6 start-page: 321 year: 1978 ident: 10.1016/j.ic.2017.04.012_br0060 article-title: Borel-programmable functions publication-title: Ann. Probab. doi: 10.1214/aop/1176995576 contributor: fullname: Blackwell – year: 2012 ident: 10.1016/j.ic.2017.04.012_br0300 contributor: fullname: Mio – volume: 286 year: 1984 ident: 10.1016/j.ic.2017.04.012_br0030 article-title: R-sets and category publication-title: Trans. Am. Math. Soc. contributor: fullname: Barua – volume: 8 issue: 4 year: 2012 ident: 10.1016/j.ic.2017.04.012_br0310 article-title: Probabilistic modal μ-calculus with independent product publication-title: Log. 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Nauk contributor: fullname: Kanoveĭ – volume: 200 start-page: 135 year: 1998 ident: 10.1016/j.ic.2017.04.012_br0350 article-title: Infinite games on finitely coloured graphs with applications to automata on infinite trees publication-title: Theor. Comput. Sci. doi: 10.1016/S0304-3975(98)00009-7 contributor: fullname: Zielonka – year: 1992 ident: 10.1016/j.ic.2017.04.012_br0340 – year: 2007 ident: 10.1016/j.ic.2017.04.012_br0050 contributor: fullname: Bertsekas – year: 1995 ident: 10.1016/j.ic.2017.04.012_br0230 contributor: fullname: Kechris – volume: 115 start-page: 97 issue: 2 year: 1983 ident: 10.1016/j.ic.2017.04.012_br0100 article-title: Classical hierarchies from a modern standpoint. II. R-sets publication-title: Fundam. Math. doi: 10.4064/fm-115-2-97-105 contributor: fullname: Burgess – volume: 81 start-page: 19 issue: 1–3 year: 2007 ident: 10.1016/j.ic.2017.04.012_br0020 article-title: Continuous separation of game languages publication-title: Fundam. 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| SubjectTerms | Computer Science Logic in Computer Science Measure theory Regular tree languages Topological complexity |
| Title | Measure properties of regular sets of trees |
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