K-theory of certain additive categories associated with varieties

Let K₀(Var_k) be the Grothendieck group of varieties over a field k. We construct an exact category, denoted Add(Var_k)_S, such that there is a surjection K₀(Vark)→K₀(Add(Var_k)_S).If we consider only zero dimensional varieties, then this surjection is an isomorphism. Like K₀(Var_k), the group K₀(Add(Var_k)_S) is also generated by isomorphism classes of varieties,and we construct motivic measures on K₀(Add(Var_k)_S) including the Euler characteristic if k=C, and point counting measures and the zeta function if k is finite.