An effective analysis of the Denjoy rank

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.