Open AccessOn Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia
Author(s) -
J. C. Ferrando,
Salvador López-Alfonso,
Manuel López Pellicer
Publication year - 2019
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil1908409f
Subject(s) - mathematics , bounded function , pointwise , infimum and supremum , norm (philosophy) , uncountable set , compact space , class (philosophy) , ring (chemistry) , discrete mathematics , banach space , combinatorics , pure mathematics , mathematical analysis , computer science , chemistry , countable set , organic chemistry , artificial intelligence , political science , law
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.