Russell G. Bilyeu

Some mapping properties of representing measures

SummaryThis paper studies relationships between operators on continuous function spaces and properties of associated vector measures given by Riesz Representation Theorems.

Uniform exhaustivity and Banach lattices

SummaryThe measure theoretic notion of strong boundedness is extended to the setting of a real σ-complete Banach lattice X; the lattice theoretic concept which corresponds to strong boundedness is termed exhaustivity. Uniformly exhaustive subsets of X are studied, and various characterizations are presented. One of these characterizations involves the order continuity of the norm on X. The exhaustive elements of certain classical lattices are cataloged. Applications are given to spaces of measures.

Norm attaining operators and norming functionals

The question of whether a countably additive measure with values in a Banach space attains the diameter of its range was unresolved. In this paper an example is given of a countably additive vector measure, taking values in a C(K) space, for which the diameter of the range is not attained. A property stronger than the attainment of the diameter, but which is possessed by many measures taking values in L-spaces, is shown to fail for infinite-dimensional measures into a space having smooth dual. As an application of the concept of norming functional (the existence of which is equivalent to the attainment of diameter), a characterization is given of the countably additive measures into space having smooth dual.