#### Vol. 4, No. 1, 2010

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Positive motivic measures are counting measures

### Jordan S. Ellenberg and Michael Larsen

Vol. 4 (2010), No. 1, 105–109
##### Abstract

A motivic measure is a ring homomorphism from the Grothendieck group of a field $K$ (with multiplication coming from the fiber product over Spec$\phantom{\rule{0.3em}{0ex}}K$) to some field. We show that if a real-valued motivic measure $\mu$ satisfies $\mu \left(\left[V\right]\right)\ge 0$ for all $K$-varieties $V$, then $\mu$ is a counting measure; that is, there exists a finite field $L$ containing $K$ such that $\mu \left(\left[V\right]\right)=|V\left(L\right)|$ for all $K$-varieties $V$.

##### Keywords
motives, motivic measures, finite fields
Primary: 14F43
Secondary: 14G15
##### Milestones
Received: 10 July 2009
Accepted: 10 August 2009
Published: 14 January 2010
##### Authors
 Jordan S. Ellenberg Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706 United States http://math.wisc.edu/~ellenber Michael Larsen Department of Mathematics Indiana University Bloomington, IN 47405 United States http://mlarsen.math.indiana.edu/~larsen/