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Some topological properties of vector measures and their integral maps

Published online by Cambridge University Press:  09 April 2009

R. Anantharaman  and
K. M. Garg
R. Anantharaman
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada.
K. M. Garg
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada.
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Abstract

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In the present paper we prove that every finite dimensional non-atomic measure ν is open and monotone (viz. ν–1 preserves connected sets) relative to the usual Fréchet-Nikodým topology on its domain and the relative topology on its range. An arbitrary finite dimensional measure is found on the other hand to be biquotient.

Given a vector measure ν, we further investigate the properties of its integral map Tν: φ → ∫ φdν defined on the set of functions φ in L1(|ν|) for which φ(s) ∈ [0,1] |ν|-almost everywhere. When ν is finite dimensional, Tν is found to be always open. In general, when Tν is open, the set of extreme points of the closed convex hull of the range of ν is proved to be closed, and when ν and Tν are both open, the range of ν in itself is closed.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 4 , June 1977 , pp. 453 - 466
Copyright
Copyright © Australian Mathematical Society 1977

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