Authors’ abstract: Let \(X\) be a real and infinite-dimensional Banach space. By \(\lambda_X\) we denote the vector space of all sequences \((\alpha_n)\) of real numbers such that \((\alpha_n x_n)\) lies inside the range of some \(X\)-valued measure with bounded variation for every null sequence \((x_n)\) in \(X\). Among other results we prove:
(i) \(\lambda_X\) is the largest normal sequence space \(\mu\) satisfying that every sequence \((x_n)\in \mu\{X\}\) lies inside the range of some \(X^{**}\)-valued measure with bounded variation, and
(ii) \(\lambda_X\) is a perfect space for which \(\ell_1\subset \lambda_X\subset\ell_2\).
We also determinate the sequence space \(\lambda_X\) when \(X\) is an \({\mathcal L}_p\)-space \((1\leq p\leq+\infty)\).