  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 where $C[a,b]$
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,
called $ACG_{\ast}[a,b]$, is a coanalytic but not Borel subclass of the space
$C[a,b]$, 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 the interval $[a,b]$ 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.
