On Grothendieck Sets

Abstract

[EN] 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 fmn g n =1 in ba(A) which is pointwise convergent on M is weakly convergent in ba(A), i. e., if there is m 2 ba (A) such that mn ( A) ! m ( A) for every A 2M then mn ! m weakly in ba(A). A subset M of an algebra of sets A is called a Nikodym set for ba(A) if each sequence fm n g n =1 in ba(A) which is pointwise bounded on M is bounded in ba (A). We prove that if S is a s-algebra of subsets of a set W which is covered by an increasing sequence fS n : n 2 Ng of subsets of S there exists p 2 N such that S p is a Grothendieck set for ba(A). This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a s-algebra S is covered by an increasing sequence fS n : n 2 Ng of subsets, there is p 2 N such that S p is a Nikodym set for ba (S). This also refines the Grothendieck result stating that for each s -algebra S the Banach space ` (S) is a Grothendieck space. Some applications to classic Banach space theory are given.