Norming sets and integration with respect to vector measures ✩

Let v be a countably additive measure defined on a measurable space (Ω, Σ) and taking values in a Banach space X. Let f : Ω → ℝ be a measurable function. In order to check the integrability (respectively, weak integrability) of f with respect to v it is sometimes enough to test on a norming set Λ ⊂ X*. In this paper we show that this is the case when A is a James boundary for BX* (respectively, Λ is weak*-thick). Some examples and applications are given as well.