Definability Aspects of the Denjoy Integral

TL;DR

This paper investigates the definability and descriptive set-theoretic complexity of the Denjoy integral and its associated function spaces, revealing their coanalytic non-Borel nature and model-theoretic properties.

Contribution

It establishes that the graph of the indefinite Denjoy integral is coanalytic non-Borel and characterizes the class of indefinite Denjoy integrals within descriptive set theory.

Findings

The graph of the indefinite Denjoy integral is coanalytic non-Borel.

The class of indefinite Denjoy integrals is coanalytic but not Borel.

The associated function spaces are elementarily equivalent and have a common decidable theory.

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 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.