Abstract
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.
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