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An effective analysis of the Denjoy rank. (English) Zbl 1484.03101

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
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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References:

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