Evaluation of motivic functions, non-nullity, and integrability in fibers

Abstract

We define an operation of evaluation at a point for motivic constructible (exponential) functions from the Cluckers-Loeser framework of motivic integration and show that two such motivic functions are abstractly equal if and only if their evaluations at each point are the same. We similarly characterise relative integrability in terms of integrability in each fiber separately. These results simplify the mentioned frameworks of motivic integration and their usage.