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A Characterization of Vector Measures with Convex Range
Author(s) -
Gouweleeuw José M.
Publication year - 1995
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-70.2.336
Subject(s) - mathematics , partition (number theory) , convexity , combinatorics , regular polygon , measure (data warehouse) , range (aeronautics) , vector space , characterization (materials science) , discrete mathematics , pure mathematics , geometry , physics , materials science , composite material , database , computer science , financial economics , optics , economics
Let μ → = ( μ 1 , … , μ n ) t be a vector measure on a measurable space (Ω, F ) such that each μ j is non‐negative and finite. In this paper, necessary and sufficient conditions on μ → are given under which the matrix‐ k ‐range MR k ( μ → ) = {( μ j ( P l ) ) j = 1l = 1 nk: ( P 1 , … , P k ) is a measurable partition of Ω} is convex. This leads to necessary and sufficient conditions on μ → under which the range R ( μ → ) = { ( μ 1 ( F ) , … , μ n( F ) ) t : F ∈ F } is convex and conditions on μ → such that the partition‐range PR ( μ → ) ( = { ( μ 1 ( P 1 ) , … , μ n( P n ) ) t : ( P 1 , … , P n ) is a measurable partition of Ω}) is convex. The results are generalizations of Lyapounov's convexity theorem. The key ingredient in the proof is a decomposition theorem for vector measures, based on the vector atoms of the measure. Applications to partitioning and classification problems are given.