On Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia

Abstract

If R is a ring of subsets of a set ? and ba (R) is the Banach space of bounded finitely additive measures defined on R equipped with the supremum norm, a subfamily ? of R is called a Nikod?m set for ba (R) if each set {?? : ???} in ba (R) which is pointwise bounded on ? is norm-bounded in ba (R). If the whole ring R is a Nikod?m set, R is said to have property (N), which means that R satisfies the Nikod?m-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck?s property (G) and prove that a ring R has property (G) if and only if the set of the evaluations on the sets of R is a so-called Rainwater set for ba(R). Recalling that R is called a (wN)-ring if each increasing web in R contains a strand consisting of Nikod?m sets, we also give a partial solution to a question raised by Valdivia by providing a class of rings without property (G) for which the relation (N)?(wN) holds.