Countably additive restrictions of vector-valued quasi measures with respect to range preservation

This paper furtherly develops the question of the existence of countably additive restrictions. The problem has already been solved for scalar charges; here we face the two dimensional case. Let m:P(X) --> R2 be a continuous quasimeasure with nonnegative components, i.e. a pair m=(m1,m2) of nonnegative strongly non atomic finitely additive set functions on the power set of X. We prove that if the range of the pair is closed (which is not always true in the finitely additive setting) then there exists an algebra A such that m is countably additive on it, strongly non atomic, and the ranges m(A) and R(m) (on the whole power set) are the same. In order to obtain this result we prove a geometric properties of zonoids in R2 labelled as Hereditarily Overlapping Boundary Property. Also we give a limit theorem, showing that the R(m) can be approximated by means of subranges m(Ak) where each Ak is a σ-algebra on which m is non atomic and countably additive