Extension of a Bounded Vector Measure with Values in a Reflexive Banach Space

Abstract

A vector measure (countable additive set function with values in a Banach space) on a field may be extended to a vector measure on the generated σ- field, under certain hypotheses. For example, the extension is established for the bounded variation case [2, 5, 8], and there are more general conditions under which the extension exists [ 1 ]. The above results have as hypotheses fairly strong boundedness conditions on the n o rm of the measure to be extended. In this paper we prove an extension theorem of the same type with a restriction on the range, supposing further that the measure is merely bounded. In fact a vector measure on a σ- field is bounded (III. 4. 5 of [3]) but it is conceivable that a vector measure on a field could be unbounded.