On vector measures with values in ℓ∞

We study some aspects of countably additive vector measures with values in l(infinity) and the Banach lattices of real-valued functions that are integrable with respect to such a vector measure. On the one hand, we prove that if W subset of l*(infinity) is a total set not containing sets equivalent to the canonical basis of l(1)(c), then there is a non-countablyadditive l(infinity)-valued map nu defined on a sigma-algebra such that the composition x* (degrees) nu is countably additive for every x* is an element of W. On the other hand, we show that a Banach lattice E is separable whenever it admits a countable, positively norming set and both E and E* are order continuous. As a consequence, if nu is a countably additive vector measure defined on a sigma-algebra and taking values in a separable Banach space, then the space L-1(nu) is separable whenever L-1(nu)* is order continuous.