Stability under integration of sums of products of real globally subanalytic functions and their logarithms

TL;DR

This paper proves that constructible functions, formed from globally subanalytic functions and their logarithms, are stable under integration and analyzes their integrability and decay properties.

Contribution

It establishes the stability of constructible functions under integration and provides new criteria and decay rate results for these functions.

Findings

Constructible functions are stable under Lebesgue integration.

Integrability conditions for constructible functions are characterized in Fubini-type settings.

Decay rates at infinity for constructible functions are established.

Abstract

We study Lebesgue integration of sums of products of globally subanalytic functions and their logarithms, called constructible functions. Our first theorem states that the class of constructible functions is stable under integration. The second theorem treats integrability conditions in Fubini-type settings, and the third result gives decay rates at infinity for constructible functions. Further, we give preparation results for constructible functions related to integrability conditions.