Weak Integration of Vector-Valued Functions

In this paper, X is a Hausdorff topological space, (μ a finite, positive, and countably additive set function on a σ-algebra β of subsets of X which includes the Borel sets, and μ^* the outer measure associated with μ (μ^*(A)=Inf{(μ(B):⊃A, B ∈ μ}).