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Measure-theoretic applications of higher Demuth’s theorem. (English) Zbl 1402.03060
Summary: We investigate measure-theoretic aspects of various notions of reducibility by applying analogs of O. Demuth’s Theorem [Commentat. Math. Univ. Carol. 29, No. 2, 233–247 (1988; Zbl 0646.03039)] in the hyperarithmetic and set-theoretic settings.
##### MSC:
 03D28 Other Turing degree structures 03D30 Other degrees and reducibilities in computability and recursion theory 03D32 Algorithmic randomness and dimension 03E15 Descriptive set theory 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets
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##### References:
 [1] BGM Laurent Bienvenu, Noam Greenberg, and Benoit Monin, \newblock Continuous higher randomness. \newblock \em preprint. · Zbl 1420.03100 [2] Chong, C. T.; Nies, Andre; Yu, Liang, Lowness of higher randomness notions, Israel J. Math., 166, 39-60, (2008) · Zbl 1153.03020 [3] Chong, C. T.; Yu, Liang, Randomness in the higher setting, J. Symb. Log., 80, 4, 1131-1148, (2015) · Zbl 1386.03046 [4] Chong, Chi Tat; Yu, Liang, Recursion theory, De Gruyter Series in Logic and its Applications 8, xiv+306 pp., (2015), De Gruyter, Berlin · Zbl 1334.03003 [5] de Leeuw, K.; Moore, E. F.; Shannon, C. E.; Shapiro, N., Computability by probabilistic machines. Automata studies, Annals of mathematics studies, no. 34, 183-212, (1956), Princeton University Press, Princeton, N. J. [6] Demuth, Osvald, Remarks on the structure of tt-degrees based on constructive measure theory, Comment. Math. Univ. Carolin., 29, 2, 233-247, (1988) · Zbl 0646.03039 [7] Demuth, O.; Ku\vcera, A., Remarks on $$1$$-genericity, semigenericity and related concepts, Comment. Math. Univ. Carolin., 28, 1, 85-94, (1987) · Zbl 0655.03029 [8] Downey, Rodney G.; Hirschfeldt, Denis R., Algorithmic randomness and complexity, Theory and Applications of Computability, xxviii+855 pp., (2010), Springer, New York · Zbl 1221.68005 [9] Feferman, S., Some applications of the notions of forcing and generic sets, Fund. Math., 56, 325-345, (1964/1965) · Zbl 0129.26401 [10] Hjorth, Greg; Nies, Andr\'e, Randomness via effective descriptive set theory, J. Lond. Math. Soc. (2), 75, 2, 495-508, (2007) · Zbl 1118.03034 [11] Jech, Thomas, Set theory, Springer Monographs in Mathematics, xiv+769 pp., (2003), Springer-Verlag, Berlin · Zbl 1007.03002 [12] Jockusch, Carl G., Jr.; Shore, Richard A., Pseudojump operators. I. The r.e. case, Trans. Amer. Math. Soc., 275, 2, 599-609, (1983) · Zbl 0514.03028 [13] Jockusch, Carl G., Jr.; Shore, Richard A., Pseudojump operators. II. Transfinite iterations, hierarchies and minimal covers, J. Symbolic Logic, 49, 4, 1205-1236, (1984) · Zbl 0574.03026 [14] Kautz, Steven M., Degrees of random sets, 129 pp., (1991), ProQuest LLC, Ann Arbor, MI [15] Kechris, Alexander S., Measure and category in effective descriptive set theory, Ann. Math. Logic, 5, 337-384, (1972/73) · Zbl 0277.02019 [16] Kjos-Hanssen, Bj\o rn; Nies, Andr\'e; Stephan, Frank; Yu, Liang, Higher Kurtz randomness, Ann. Pure Appl. Logic, 161, 10, 1280-1290, (2010) · Zbl 1223.03025 [17] Kurtz, Stuart Alan, Randomness and Genericity in the Degrees of Unsolvability, 138 pp., (1981), ProQuest LLC, Ann Arbor, MI [18] Lerman, Manuel, Degrees of unsolvability, Perspectives in Mathematical Logic, xiii+307 pp., (1983), Springer-Verlag, Berlin · Zbl 1365.03002 [19] Martin-L\"of, Per, On the notion of randomness. Intuitionism and Proof Theory, Proc. Conf., Buffalo, N.Y., 1968, 73-78, (1970), North-Holland, Amsterdam [20] Miller, Joseph S.; Yu, Liang, On initial segment complexity and degrees of randomness, Trans. Amer. Math. Soc., 360, 6, 3193-3210, (2008) · Zbl 1140.68028 [21] Nies, Andr\'e, Computability and randomness, Oxford Logic Guides 51, xvi+433 pp., (2009), Oxford University Press, Oxford · Zbl 1237.03027 [22] Odifreddi, Piergiorgio, Classical recursion theory, Studies in Logic and the Foundations of Mathematics 125, xviii+668 pp., (1989), North-Holland Publishing Co., Amsterdam · Zbl 0661.03029 [23] Posner, David B.; Robinson, Robert W., Degrees joining to $$\textbf{0}^{′ }$$, J. Symbolic Logic, 46, 4, 714-722, (1981) · Zbl 0517.03014 [24] Sacks, Gerald E., Degrees of unsolvability, ix+174 pp., (1963), Princeton University Press, Princeton, N.J. [25] Sacks, Gerald E., Measure-theoretic uniformity in recursion theory and set theory, Trans. Amer. Math. Soc., 142, 381-420, (1969) · Zbl 0209.01603 [26] Sacks, Gerald E., Higher recursion theory, Perspectives in Mathematical Logic, xvi+344 pp., (1990), Springer-Verlag, Berlin · Zbl 0699.00011 [27] Soare, Robert I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, xviii+437 pp., (1987), Springer-Verlag, Berlin · Zbl 0623.03042 [28] Solovay, Robert M., A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2), 92, 1-56, (1970) · Zbl 0207.00905 [29] Spector, Clifford, Measure-theoretic construction of incomparable hyperdegrees, J. Symb. Logic, 23, 280-288, (1958) · Zbl 0085.24901 [30] Wang, Wei, Relative enumerability and 1-genericity, J. Symbolic Logic, 76, 3, 897-913, (2011) · Zbl 1260.03079 [31] Yu, Liang, Lowness for genericity, Arch. Math. Logic, 45, 2, 233-238, (2006) · Zbl 1148.03033 [32] Yu, Liang, Measure theory aspects of locally countable orderings, J. Symbolic Logic, 71, 3, 958-968, (2006) · Zbl 1109.03038
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