2 00 3 What Is Motivic Measure ?

This article gives an exposition of the theory of arithmetic motivic measure, as developed by J. Denef and F. Loeser. 1. Preliminary Concepts There is much that is odd about motivic measure if it is judged by measure theory in the sense of twentieth century analysis. It does not fit neatly with the tradition of measure in the style of Hausdorff, Haar, and Lebesgue. It is best to view motivic measure as something new and different, and to recognize that when it comes to motivic measure, the term ‘measure’ is used loosely. Motivic measure will be easier to understand, once two of its peculiarities are explained. The first peculiarity is that the measure is not real-valued. Rather, it takes values in a scissor group. An introductory section on scissor groups for polygons will recall the basic facts about these groups. The second peculiarity is that rather than a boolean algebra of measurable sets, we work directly with the underlying boolean formulas that define the sets. The reasons for working directly with boolean formulas will be described in a second introductory section. After these two introductory remarks, we will describe ‘motivic counting’ in Section 2. Motivic counting is to ordinary counting what motivic measure is to ordinary measure. Motivic counting will lead into motivic measure. 1.1. Scissor Groups for polygons. Motivic volume is defined by a process that is similar to the scissor-group construction of the area of polygons in the plane. To draw out the similarities, let us recall the construction. It determines the area of a polygons without taking limits. Any polygon in the plane can be cut into finitely many triangles that can be reassembled into a rectangle of unit width. Figure 1 illustrates three steps (2, 3, and 4) of the general algorithm. The algorithm consists of 5 elementary transformations. (1) Triangulate the polygon. (2) Transform triangles into rectangles. (3) Fold long rectangles in half. (4) Rescale each rectangle to give it an edge of unit width. (5) Stack all the unit width rectangles end to end. The length of the unit width rectangle is the area. An abelian group encodes these cut and paste operations. Let F be the free abelian group on the set of polygons in the plane. We impose two families of relations: 1