Definability aspects of the Denjoy integral

Sean Walsh¹

¹ Department of Logic and Philosophy of Science University of California, Irvine 5100 Social Science Plaza Irvine, CA 92697-5100, U.S.A.

Fundamenta Mathematicae, Tome 237 (2017) no. 1, pp. 1-29.

Résumé

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.

Détail

DOI : 10.4064/fm773-9-2016

Keywords: denjoy integral integral extends lebesgue integral integrate derivative paper shown graph indefinite denjoy integral mapsto int coanalytic non borel relation product space times where polish space real valued measurable functions polish space real valued continuous functions using methods shown class indefinite denjoy integrals denoted acg ast coanalytic borel subclass answering question posed dougherty kechris basic model theory associated spaces integrable functions studied here main result viewed mathbb module indeterminate being interpreted indefinite integral space continuous functions elementarily equivalent lebesgue integrable denjoy integrable functions interval each stable superstable have common decidable theory viewed mathbb modules