How to Obtain Maximal and Minimal Subranges of Two-Dimensional Vector Measures

Let (X, ℱ) be a measurable space with a nonatomic vector measure µ =(µ₁, µ₂). Denote by R(Y) the subrange R(Y)= {µ(Z): Z ∈ ℱ, Z ⊆ Y }. For a given p ∈ µ(ℱ) consider a family of measurable subsets ℱ_p = {Z ∈ ℱ : µ(Z)= p}. Dai and Feinberg proved the existence of a maximal subset Z* ∈ F_p having the maximal subrange R(Z*) and also a minimal subset M* ∈ 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 Dai and Feinberg.