Positive motivic measures are counting measures

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$K$) 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 $\mu \left(\left[V\right]\right)=\left|V\left(L\right)\right|$ for all $K$-varieties $V$.