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Lyapunov convexity type theorems for non-atomic vector measures
Abstract
Given a Banach space X, we consider the problem
of when the range of a non-atomic, and σ-additive X-valued measure has a
convex closure. We give a survey of Lyapunov convexity type theorems pertaining
to this problem. We also give a necessary and sufficient condition that will
insure the convexity of the closure of the range of the measure. In particular,
we show that the closure of the range of a non-atomic and σ-additive X-valued
measure is convex whenever it is compact.
Mathematics Subject Classification (2000): 46G10, 46B99
Key words: Vector measures, Lyapunov convexity theorem, Lyapunov property, Banach spaces
Quaestiones Mathematicae 26(2003), 371–383
of when the range of a non-atomic, and σ-additive X-valued measure has a
convex closure. We give a survey of Lyapunov convexity type theorems pertaining
to this problem. We also give a necessary and sufficient condition that will
insure the convexity of the closure of the range of the measure. In particular,
we show that the closure of the range of a non-atomic and σ-additive X-valued
measure is convex whenever it is compact.
Mathematics Subject Classification (2000): 46G10, 46B99
Key words: Vector measures, Lyapunov convexity theorem, Lyapunov property, Banach spaces
Quaestiones Mathematicae 26(2003), 371–383
