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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.
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
Full Text: DOI arXiv Euclid
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