An effective analysis of the Denjoy rank

TL;DR

This paper investigates the descriptive complexity of various Denjoy-related function classes, establishing their completeness levels and providing precise classifications for sets of functions based on transfinite Denjoy totalization steps.

Contribution

It proves that several Denjoy function classes are a1^1_1-complete and determines the exact descriptive complexity of functions obtained after transfinite Denjoy totalization.

Findings

VBG, VBG_, ACG, and ACG_ are a1^1_1-complete.

The set of functions with at most  steps of Denjoy totalization is a0_{2}-complete.

Certain sets related to Denjoy integrability are a1^1_1-complete.

Abstract

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.