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We prove that the compact subsets of a Banach space X lie inside ranges of X-valued measures if and only if X* can be embedded in an LI space. In these spaces we prove that every compact is, in fact, a subset of a compact range. We also prove that if every compact of X is a subset of the range of an X-valued measure of bounded variation, then X is finite dimensional. Thus we answer a question by R. Anantharaman and J. Diestel.

Banach spaces in which every compact lies inside the range of a vector measure