Sequential closures of σ-subalgebras for a vector measure

Let X be a locally convex space, m : Σ → X be a vector measure defined on a σ-algebra Σ, and L1(m) be the associated (locally convex) space of m-integrable functions. Let Σ(m) denote {χ E ;E ∈ Σ}, equipped with the relative topology from L1(m). For a subalgebra A ⊆ Σ, let Aσ denote the generated σ-algebra and As denote the sequential closure of χ(A) = {χ E ;E ∈ A} in L1(m). Sets of the form As arise in criteria determining separability of L1(m); see [6]. We consider some natural questions concerning As and, in particular, its relation to χ(Aσ). It is shown that As ⊆ Σ(m) and moreover, that {E ∈ Σ;χ E ∈ As} is always a σ-algebra and contains Aσ . Some properties of X are determined which ensure that χ(Aσ) = As, for any X-valued measure m and subalgebra A ⊆ Σ; the class of such spaces X turns out to be quite extensive.