Positive motivic measures are counting measures

TL;DR

This paper proves that the only positive motivic measures on the Grothendieck ring of varieties over a field are counting measures based on rational points over finite extensions, establishing a uniqueness result.

Contribution

It demonstrates that positive motivic measures are exclusively counting measures, clarifying their structure and limiting possible measures to rational point counts over finite fields.

Findings

Positive motivic measures are only counting measures.

Counting measures correspond to rational points over finite field extensions.

Uniqueness of positive motivic measures established.

Abstract

Let K be a field. A positive motivic measure on the Grothendieck ring K_0(Var_K) is a homomorphism from K_0(Var_K) to the real numbers assigning a nonnegative value to every variety. In this note we show that the only positive motivic measures are the counting measures: measures on K_0(Var_{F_q}) which send a variety to its number of rational points over some fixed finite extension of F_q.