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 wellfounded 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1711.00154
 arXiv:
 arXiv:1711.00154
 Bibcode:
 2017arXiv171100154B
 Keywords:

 Mathematics  Logic;
 Mathematics  Classical Analysis and ODEs
 EPrint:
 17 pages