K-theory of certain additive categories associated with varieties

<div>Let <i>K₀</i>(Var<i>_k</i>) be the Grothendieck group of varieties over a field <i>k</i>. We construct an exact category, denoted Add(Var_<i>k</i>)_<i>S</i>, such that there is a surjection <i>K₀</i>(Var<i>k</i>)→<i>K₀</i>(Add(Var<i>_k</i>)_<i>S</i>).If we consider only zero dimensional varieties, then this surjection is an isomorphism. Like <i>K₀</i>(Var<i>_k</i>), the group K_<i>0</i>(Add(Var_<i>k</i>)<i>_S</i>) is also generated by isomorphism classes of varieties,and we construct motivic measures on <i>K₀</i>(Add(Var<i>_k</i>)<i>_S</i>) including the Euler characteristic if <i>k</i>=<i>C</i>, and point counting measures and the zeta function if <i>k</i> is finite.<br></div>