Absolute continuity and on the range of a vector measure

by De Kock, Mienie

Let ? be a compact Hausdorff space1 with Borel ?-field ? and let ? and ? be regular Borel probabilities on ?. Then the following are equivalent: (a) [C(?) ? L^1(?)] (b) ? (c) [B(?) ? L^1(?)] where B(?) is the Banach space of all bounded Borel measurable functions equipped with the supremum norm. We extend this result to vector-valued cases. 2. To which Banach spaces X is it so that if C is a countable subset of X that lies in the range of a countably additive X??-valued measure with the same ?-field domain, then there is an X-valued countably additive measure with a ?-field domain, whose range also contains C? We give a partial solution to the problem.