Some Applications of Grothendieck Sets

We call a subset M of an algebra of sets A a Grothendieck set for the Banach space ba(A) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence {μn}∞ n=1 in ba(A) which is pointwise convergent onMis weakly convergent in ba(A), i. e., if there is μ ∈ ba (A) such that μn (A) → μ (A) for every A ∈ M then μn → μ weakly in ba(A). A subset M of an algebra of sets A is called a Nikodym set for ba(A) if each sequence {μn}∞ n=1 in ba(A) which is pointwise bounded on M is bounded in ba(A). We prove that if Σ is a σ-algebra of subsets of a set Ω which is covered by an increasing sequence {Σn : n ∈ N} of subsets of Σ there exists p ∈ N such that Σp is a Grothendieck set for ba(Σ). This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a σ-algebra Σ is covered by an increasing sequence {Σn : n ∈ N} of subsets, there is p ∈ N such that Σp is a Nikod´ym set for ba (Σ). This also refines the Grothendieck result stating that for each σ-algebra Σ the Banach space ℓ∞ (Σ) is a Grothendieck space. Some applications to classic Banach space theory are given.