Commutative sequences of integrable functions and best approximation with respect to the weighted vector measure distance

Let lambda be a countably additive vector measure with values in a separable real Hilbert space H. We define and study a pseudo metric on a Banach lattice of integrable functions related to lambda that we call a lambda-weighted distance. We compute the best approximation with respect to this distance to elements of the function space by the use of sequences with special geometric properties. The requirements on the sequence of functions axe given in terms of a commutation relation between these functions that involves integration with respect to lambda. We also compare the approximation that is obtained in this way with the corresponding projection on a particular Hilbert space.