Igor Kluvánek

The range of a vector-valued measure

It has been known, since 1940, that the range of a non-atomic finitedimensional space-valued measure on a a-algebra is a compact convex set (Liapunov [7]). But this fails to be true, in general, for measures with values in an arbitrary locally convex topological vector space, even a Banach space. In fact, examples have been constructed o f non-atomic measures with closed but non-convex and non-compact range and even with non-closed non-convex range (see, e.g., Uhl [10]). It is proved in the first section o f this note that the weak closure o f the range o f such measure is convex if the range-space is assumed metrizable or if the a-algebra serving for the domain o f the measure is essentially countably generated, but, in general, even the weak closure o f the range does not have to be convex. The second and third sections are devoted to the problem of expressing every member o f the closed convex hull o f the range as the integral of a function with values in [0, 1]. Solution of this problem gives conditions on whether or not every extremal point of the closed convex hull o f the range belongs to the range. In the final section the results are generalized slightly in order to show the relevance o f these results to the uniqueness questions in Linear Time Optimal Control Theory.

THE EXTENSION AND CLOSURE OF VECTOR MEASURE

Publisher Summary This chapter discusses the extension and closure of vector measures. The story of extension usually does not start with a ring. One often needs to construct a measure defined originally on a semi-ring or even on a lattice of sets. The passage from a semi-ring to the ring or from a lattice to a semi-ring is usually algebraic in character. Every additive function is extendable from a semi-ring onto the ring, and the σ-additivity carries over. The property of a measure on a semi-ring guarantees extendability onto the σ-ring. The theory of extension, closure, and weak compactness of vector measures has an analogy in a similar theory for vector Daniell integrals.

Characterization of the closed convex hull of the range of a vector-valued measure

Abstract For a set K in a locally convex topological vector space X there exists a set T, a σ-algebra S of subsets of T and a σ-additive measure m: S → X such that K is the closed convex hull of the range {m(E): E ∈ S } of the measure m if and only if there exists a conical measure u on X so that K Ku,Ku, the set of resultants of all conical measures v on X such that v

The range of a vector measure

Let X be a real quasi-complete locally convex topological vector space. Let K C X be a weakly compact convex and symmetric set such that 0 GK. Let T be an abstract space and 5 be a a-algebra of subsets of T. A vector measure is a a-additive mapping m: S —* X. We are concerned with the question whether there exists a vector measure m: S ~ * X such that K coincides with the closed convex hull of the range of m, i.e. K = œ m(S) = co{m(E): E G S}. The case X = R was surveyed in [1] .