HOW TO OBTAIN MAXIMAL AND MINIMAL SUBRANGES OF TWO-DIMENSIONAL VECTOR MEASURES

Abstract

Let $ (X, \mathcal{F}) $ be a measurable space with a nonatomic vector measure $ \mu=(\mu_{1},\mu_{2}) $. Denote by $ R(Y) $ the subrange $R(Y)=\lbrace \mu(Z): Z \in \mathcal{F}, Z \subseteq Y \rbrace $. For a given $ p \in \mu(\mathcal{F}) $ consider a family of measurable subsets $ \mathcal{F}_{p} = \lbrace Z \in \mathcal{F}: \mu(Z)=p \rbrace. $ Dan and Feinberg proved the existence of a maximal subset $ Z^{*} \in \mathcal{F}_{p} $ having the maximal subrange $ R(Z^{*})$ and also a minimal subset $ M^{*} \in \mathcal{F}_{p} $ with the minimal subrange $ R(M^{*})$. We present a method of obtaining the maximal and the minimal subsets. Hence, we get simple proofs of the results of Dan and Feinberg.