Vector measure Maurey–Rosenthal-type factorizations and -sums of -spaces

Consider a vector measure of bounded variation m with values in a Banach space and an operator T : X −→ L1(m), where L1(m) is the space of integrable functions with respect to m. We characterize when T can be factorized through the space L2(m) by means of a multiplication operator given by a function of L2(|m|), where |m| is the variation of m, extending in this way the Maurey–Rosenthal Theorem. We use this result to obtain information about the structure of the space L1(m) when m is a sequential vector measure. In this case the space L1(m) is an -sum of L1-spaces.