TY - JOUR
T1 - Decomposition of ranges of vector measures
AU - Neyman, Abraham
PY - 1981/3
Y1 - 1981/3
N2 - The following conditions on a zonoid Z, 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 zonoid Z determines the m-range of any non-atomic vector measure with range Z, where the m-range of a vector measure μ is the set of m-tuples (μ(S 1), ..., μ(S m), where S 1, ...S m are disjoint measurable sets and (iii) there is avector measure space (X, Σ, μ) such that any finite factorization of Z, Z =ΣZ i, 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.
AB - The following conditions on a zonoid Z, 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 zonoid Z determines the m-range of any non-atomic vector measure with range Z, where the m-range of a vector measure μ is the set of m-tuples (μ(S 1), ..., μ(S m), where S 1, ...S m are disjoint measurable sets and (iii) there is avector measure space (X, Σ, μ) such that any finite factorization of Z, Z =ΣZ i, 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.
UR - http://www.scopus.com/inward/record.url?scp=51249179638&partnerID=8YFLogxK
U2 - 10.1007/BF02761817
DO - 10.1007/BF02761817
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AN - SCOPUS:51249179638
SN - 0021-2172
VL - 40
SP - 54
EP - 64
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -