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An effective analysis of the Denjoy rank. (English) Zbl 07222690
Summary: We analyze the descriptive complexity of several \(\Pi^1_1\)-ranks from classical analysis which are associated to Denjoy integration. We show that \(VBG, VBG_{\ast}, ACG\), and \(ACG_{\ast}\) are \(\Pi^1_1\)-complete, answering a question of Walsh in case of \(ACG_{\ast}\). Furthermore, we identify the precise descriptive complexity of the set of functions obtainable with at most \(\alpha\) steps of the transfinite process of Denjoy totalization: if \(|\cdot|\) is the \(\Pi^1_1\)-rank naturally associated to \(VBG, VBG_{\ast}\), or \(ACG_{\ast}\), and if \(\alpha < \omega_1^{ck}\), then \(\{F \in C(I) : |F| \leq \alpha\}\) is \(\Sigma^0_{2\alpha}\)-complete. These finer results are an application of the author’s previous work on the limsup rank on well-founded trees. Finally, \(\{(f, F) \in M(I) \times C(I) : F \in ACG_{\ast} \text{ and } F' = f \text{ a.e.}\}\) and \(\{f \in M(I) : f \text{ is Denjoy integrable}\}\) are \(\Pi^1_1\)-complete, answering more questions of Walsh.
MSC:
03E15 Descriptive set theory
26A39 Denjoy and Perron integrals, other special integrals
03D45 Theory of numerations, effectively presented structures
03C57 Computable structure theory, computable model theory
03D25 Recursively (computably) enumerable sets and degrees
03D30 Other degrees and reducibilities in computability and recursion theory
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
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References:
[1] Ash, C. J., and J. Knight, Computable Structures and the Hyperarithmetical Hierarchy, vol. 144 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 2000. · Zbl 0960.03001
[2] Cenzer, D., and R. D. Mauldin, “On the Borel class of the derived set operator,” Bulletin de la Société Mathématique de France, vol. 110 (1982), pp. 357-80. · Zbl 0514.54027
[3] Cenzer, D., and R. D. Mauldin, “On the Borel class of the derived set operator, II,” Bulletin de la Société Mathématique de France, vol. 111 (1983), pp. 367-72. · Zbl 0552.54027
[4] Dougherty, R., and A. S. Kechris, “The complexity of antidifferentiation,” Advances in Mathematics, vol. 88 (1991), pp. 145-69. · Zbl 0739.26005
[5] Greenberg, N., A. Montalbán, and T. A. Slaman, “Relative to any non-hyperarithmetic set,” Journal of Mathematical Logic, vol. 13 (2013), art. ID 1250007. · Zbl 1308.03050
[6] Holický, P., S. P. Ponomarev, L. Zajíček, and M. Zelený, “Structure of the set of continuous functions with Luzin’s property (N),” Real Analysis Exchange, vol. 24 (1998/99), pp. 635-56. · Zbl 0968.26008
[7] Kechris, A. S., and W. H. Woodin, “Ranks of differentiable functions,” Mathematika, vol. 33 (1986), pp. 252-78. · Zbl 0618.03024
[8] Lempp, S., “Hyperarithmetical index sets in recursion theory,” Transactions of the American Mathematical Society, vol. 303 (1987), pp. 559-83. · Zbl 0652.03030
[9] Saks, S., Theory of the Integral, 2nd revised edition, Dover, New York, 1964. · Zbl 1196.28001
[10] Walsh, S., “Definability aspects of the Denjoy integral,” Fundamenta Mathematicae, vol. 237 (2017), pp. 1-29. · Zbl 1420.03086
[11] Westrick, L. · Zbl 1338.03089
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