On the product of vector measures

Extract

Let μ and ν be measures defined on some σ-algebrs with values in locally convex topological vector spaces X and Y, repectively. It is possible [1] to construct their product λ = μ × ν as a measure on a σ-algebra if λ is allowed to take its values in X ⊗_ε Y, the completion of X ⊗ Y in the topology of bi-equicontinuous convergence. The reason is, roughly speaking, that the topology of biequicontinuous convergence on X ⊗ Y is coarse enough to make λ σ-additive and the completion X ⊗_ε Y is big enough to accommodate all values of λ. Here we are going to improve the result by introducing a finer topology on X ⊗ Y in which λ will be σ-additive and such that all values of λ will belong to the completion of X ⊗ Y under that topology. The topology in question is obtained by a slight modification from a topology considered for the first time in the work [3] of Jacobs. Curiously enough, the proof of the improved result is simpler than that of [1] and reduces almost to a direct observation avoiding duality arguments.