Notre Dame Journal of Formal Logic

An Effective Analysis of the Denjoy Rank

Linda Westrick

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We analyze the descriptive complexity of several Π11-ranks from classical analysis which are associated to Denjoy integration. We show that VBG, VBG*, ACG, and ACG* are Π11-complete, answering a question of Walsh in case of ACG*. Furthermore, we identify the precise descriptive complexity of the set of functions obtainable with at most α steps of the transfinite process of Denjoy totalization: if || is the Π11-rank naturally associated to VBG, VBG*, or ACG*, and if α<ω1ck, then {FC(I):|F|α} is Σ2α0-complete. These finer results are an application of the author’s previous work on the limsup rank on well-founded trees. Finally, {(f,F)M(I)×C(I):FACG*andF'=fa.e.} and {fM(I):fis Denjoy integrable} are Π11-complete, answering more questions of Walsh.

Article information

Notre Dame J. Formal Logic, Volume 61, Number 2 (2020), 245-263.

Received: 31 October 2017
Accepted: 4 November 2019
First available in Project Euclid: 4 April 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E15: Descriptive set theory [See also 28A05, 54H05]
Secondary: 26A39: Denjoy and Perron integrals, other special integrals

Denjoy totalization coanalytic ranks


Westrick, Linda. An Effective Analysis of the Denjoy Rank. Notre Dame J. Formal Logic 61 (2020), no. 2, 245--263. doi:10.1215/00294527-2020-0006.

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