Tensor products of vector measures and sequences in the range of a vector measure

We characterize those Banach spaces $X$, in which every $X$-valued measure with relatively compact range admits product with any vector measure and with respect to any bilinear map, as those $X$ such that $\Pi _{1} (X,\ell _{1}) = {\mathcal {L}} (X,\ell _{1})$. We also show that this condition is equivalent to the condition that every sequence in $X$ that lies inside the range of a measure with relatively compact range, actually lies inside the range of a measure of bounded variation.