# K-theory of certain additive categories associated with varieties

Let

*K*(Var_{0}*) be the Grothendieck group of varieties over a field*_{k}*k*. We construct an exact category, denoted Add(Var_{k})_{S}, such that there is a surjection*K*(Var_{0}*k*)→*K*(Add(Var_{0}*)*_{k}_{S}).If we consider only zero dimensional varieties, then this surjection is an isomorphism. Like*K*(Var_{0}*), the group K*_{k}_{0}(Add(Var_{k})*) is also generated by isomorphism classes of varieties,and we construct motivic measures on*_{S}*K*(Add(Var_{0}*)*_{k}*) including the Euler characteristic if*_{S}*k*=*C*, and point counting measures and the zeta function if*k*is finite.