We analyze the descriptive complexity of several ∏1 1-ranks from classical analysis which are associated to Denjoy integration. We show that VBG, VBG*, ACG, and ACG* are ∏1 1-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 ∣·∣ j is the ∏1 1-rank naturally associated to VBG, VBG*, or ACG*, and if (equation presented)-complete. These finer results are an application of the author's previous work on the limsup rank on well-founded trees. Finally, (equation presented) is Denjoy integrableºare ∏1 1-complete, answering more questions of Walsh.
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