Abstract

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<ωck1, then {F∈C(I):|F|≤α} is Σ02α-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):F∈ACG∗andF'=fa.e.} and {f∈M(I):fis Denjoy integrable} are Π11-complete, answering more questions of Walsh.