Decomposition of ranges of vector measures

Abstract

The following conditions on a zonoidZ, i.e., a range of a non-atomic vector measure, are equivalent: (i) the extreme set containing 0 in its relative interior is a parallelepiped; (ii) the zonoidZ determines them-range of any non-atomic vector measure with rangeZ, where them-range of a vector measure μ is the set ofm-tuples (μ(S1), …, μ(Sm), whereS1, …Smare disjoint measurable sets and (iii) there is avector measure space (X, Σ, μ) such that any finite factorization ofZ, Z =ΣZi, in the class of zonoids could be achieved by decomposing (X, Σ). In the case of ranges of non-atomic probability measures (i) is automatically satisfied, so (ii) and (iii) hold.