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 $\mathit{VBG}$, ${\mathit{VBG}}_{*}$, $\mathit{ACG}$, and ${\mathit{ACG}}_{*}$ are ${\Pi }_1^1$-complete, answering a question of Walsh in case of ${\mathit{ACG}}_{*}$. 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 $\mathit{VBG}$, ${\mathit{VBG}}_{*}$, or ${\mathit{ACG}}_{*}$, and if $\alpha <{\omega }_1^{ck}$, then $\{F\in C(I):|F|\le \alpha \}$ is ${\Sigma }_{2\alpha }^0$-complete. These finer results are an application of the author’s previous work on the limsup rank on well-founded trees. Finally, $\left\{\right(f,F)\in M(I)\times C(I):F\in {\mathit{ACG}}_{*}\text{and}F\text{'}=f\text{a.e.}\}$ and $\{f\in M(I):f\text{is Denjoy integrable}\}$ are ${\Pi }_1^1$-complete, answering more questions of Walsh.