## Notre Dame Journal of Formal Logic

### An Effective Analysis of the Denjoy Rank

Linda Westrick

#### Abstract

We analyze the descriptive complexity of several $\Pi^{1}_{1}$-ranks from classical analysis which are associated to Denjoy integration. We show that $\mathit{VBG}$, $\mathit{VBG}_{\ast}$, $\mathit{ACG}$, and $\mathit{ACG}_{\ast}$ are $\Pi^{1}_{1}$-complete, answering a question of Walsh in case of $\mathit{ACG}_{\ast}$. 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 $\mathit{VBG}$, $\mathit{VBG}_{\ast}$, or $\mathit{ACG}_{\ast}$, and if $\alpha\lt \omega_{1}^{ck}$, 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, $\{(f,F)\in M(I)\times C(I):F\in\mathit{ACG}_{\ast}\text{ and }F'=f\text{ a.e.}\}$ and $\{f\in M(I):f\text{ is Denjoy integrable}\}$ are $\Pi^{1}_{1}$-complete, answering more questions of Walsh.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1585965655

Digital Object Identifier
doi:10.1215/00294527-2020-0006

Mathematical Reviews number (MathSciNet)
MR4092534

Zentralblatt MATH identifier
07222690

#### Citation

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. https://projecteuclid.org/euclid.ndjfl/1585965655

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