Loci of Integrability, Zero Loci, and Stability Under Integration for Constructible Functions on Euclidean Space with Lebesgue Measure

Abstract

We introduce and study loci of integrability. We prove a correspondence between zero loci and loci of integrability for constructible functions on Euclidean space, where a function is called constructible if it is a sum of products of globally subanalytic functions and of logarithms of globally subanalytic functions. We generalize the main result of the authors in Cluckers and Miller [Stability under integration of sums of products of real globally subanalytic functions and their logarithms. Duke Mathematical Journal 156, no. 2 (2011)] about the stability under integration of the class of constructible functions, by relaxing the conditions on integrability. Further, we give an interpolation result for constructible functions by constructible functions with maximal locus of integrability.