Bounded integral and motivic Milnor fiber

Abstract: We construct a new motivic integration morphism, the so-call bounded integral, that interpolates both the integration morphisms with and without volume forms of Hrushovski and Kazhdan. This is done within the framework of model theory of algebraically closed valued fields of equicharacteristic zero. As an application, we recover and extend some results of Hrushovski and Loeser about the motivic Milnor fiber.