## Definability aspects of the Denjoy integral

### Volume 237 / 2017

#### Abstract

The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this paper, it is shown that the graph of the indefinite Denjoy integral $f \mapsto \int _a^x f$ is a coanalytic non-Borel relation on the product space $M[a,b]\times C[a,b]$, where $M[a,b]$ is the Polish space of real-valued measurable functions on $[a,b]$, and $C[a,b]$ is the Polish space of real-valued continuous functions on $[a,b]$. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, denoted by $ACG_{\ast }[a,b]$, is a coanalytic but not Borel subclass of $C[a,b]$, thus answering a question posed by Dougherty and Kechris. Some basic model theory of the associated spaces of integrable functions is also studied. Here the main result is that, when viewed as an $\mathbb {R}[X]$-module with the indeterminate $X$ being interpreted as the indefinite integral, the space of continuous functions on $[a,b]$ is elementarily equivalent to the Lebesgue integrable and Denjoy integrable functions on this interval, and each is stable but not superstable, and that they all have a common decidable theory when viewed as $\mathbb {Q}[X]$-modules.