Definability of the Integrability Locus in Polynomially Bounded o-Minimal Structures

TL;DR

This paper demonstrates that in polynomially bounded o-minimal structures, the set of parameters leading to finite-volume fibers in definable families is itself definable, consolidating scattered arguments into a unified result.

Contribution

It provides a unified proof that the set of parameters with finite-volume fibers is definable in polynomially bounded o-minimal structures, clarifying previous scattered arguments.

Findings

The set of parameters with finite-volume fibers is definable.

The proof consolidates scattered arguments in the literature.

Applicable to polynomially bounded o-minimal structures.

Abstract

A note connecting arguments scattered in the extant literature proving that, in any o-minimal expansion of the real field, a definable family of sets has the property that the set of parameters corresponding to finite-volume fibers is definable.