ABSTRACT. These notes grew out of the authors effort to understand the theory of motivic integration. They give a short but thorough introduction to the flavor of motivic integration which nowadays goes by the name of geometric motivic integration. Motivic inegration was introduced by Kontsevich and the foundations were worked out by Denef, Loeser, Batyrev and Looijenga. We focus on the smooth complex case and present the theory as self contained as possible. As an illustration we give some applications to birational geometry which originated in the work of Musta¸tǎ. CONTENTS 1. The invention of motivic integration. 2 2. Geometric motivic integration 4 2.1. The value ring of the motivic measure
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