Adm)atrdd hence the asseruon. let am be the system of all finite disjoint unions of sets of the form ci × × cm, with c1, , cm e x. this is a field: see prop. 3.2.3 in [30]. from the first step of the proof and additivity of ? (m) we deduce that ? (m)(a) 0 holds for all a e am. the system m of all sets a with the property that um(a) 2 0 is closed with respect to (countable) monotone unions and intersections. hence theorem a.2 implies that m x'". therefore ? (m) is non-negative lemma a.17 let ? , v e n