The range of a vector measure determines its total variation

Abstract:

We prove that if the ranges of two finitely additive measures with values in a normed space have the same closed convex hull, then the measures have the same total variation. We also study the monotonicity of this variation with respect to the range, proving that a normed space $X$ is $C$-isomorphic to a subspace of an ${L^1}$ space if and only if, for every pair $\mu ,\nu$ of $X$-valued measures such that the range of $\mu$ lies inside the closed convex hull of the range of $\nu$, the total variation of $\mu$ is less than or equal to $C$ times the total variation of $\nu$. This allows us to answer two questions raised by R. Anantharaman and J. Diestel.