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A New Concept for a Choquet Ordering
Author(s) -
Roth Walter
Publication year - 1986
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-34.1.81
Subject(s) - mathematics , choquet theory , uniqueness , choquet integral , unit sphere , pure mathematics , compact space , measure (data warehouse) , regular polygon , norm (philosophy) , unit interval , radon measure , convex set , discrete mathematics , locally compact space , mathematical analysis , convex optimization , computer science , artificial intelligence , geometry , fuzzy logic , database , political science , law
In classical Choquet theory for representing measures on compact sets the complex case is reduced to the real one by treating the real and imaginary parts of functions and measures separately. In order to obtain norm‐preservation or related stability results one has to choose special representations on compact convex subsets of the complex measure space. We offer a different approach to the complex theory. Given a linear space L of continuous complex‐valued functions on a compact set X we establish a Choquet ordering on the entire unit ball of the complex Radon measures. The ordering is defined using L ‐subharmonic functions on X × Γ (where Γ denotes the unit circle in C) which arise as canonical generalizations of L ‐subharmonic functions on X in the real case. Although when 1 ε L our notion of maximality coincides with the classical one, there is a closer relation between measures and their maximal counterparts, which implies the stability results mentioned above. There is a different approach to the uniqueness problem as well.