aequationes mathematicae

, Volume 20, Issue 1, pp 252–262 | Cite as

The Choquet integral representation in the affine vector-valued case

  • Paulette Saab
Research Papers

Abstract

LetX be any compact convex subset of a locally convex Hausdorff space andE be a complex Banach space. We denote byA(X, E) the space of all continuous and affineE-valued functions defined onX. In this paper we prove thatX is a Choquet simplex if and only if the dual ofA(X, E) is isometrically isomorphic by a selection map toM m (X, E*), the space ofE*-valued,w*-regular boundary measures onX. This extends and strengthens a result of G. M. Ustinov. To do this we show that for any compact convex setX, each element of the dual ofA(X, E) can be represented by a measure inM m (X, E*) with the same norm, and this representation is unique if and only ifX is a Choquet simplex. We also prove that ifX is metrizable andE is separable then there exists a selection map from the unit ball of the dual ofA(X, E) into the unit ball ofM m (X, E*) which is weak* to weak*-Borel measurable.

AMS (1970) subject classification

Primary 46A55 46E40 

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Copyright information

© Birkhäuser Verlag 1980

Authors and Affiliations

  • Paulette Saab
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaU.S.A.

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