On Nikodym and Rainwater sets for ba(R) and a problem of M. Valdivia
Abstract
If R is a ring of subsets of a set Ω and ba(R) is the Banach space of bounded
finitely additive measures defined on R equipped with the supremum norm, a subfamily Δ of R is called a Nikodým set for ba (R) if each set {μ_{alpha} : alpha ∈ Λ} in ba(R) which is pointwise bounded on Δ is norm-bounded in ba (R). If the whole ring R is a Nikodym set, R is said to have property (N), which means that R satisfies the Nikodym-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck’s property (G) and we 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 Nikodym 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 properties (N) and (wN) are equivalents.
finitely additive measures defined on R equipped with the supremum norm, a subfamily Δ of R is called a Nikodým set for ba (R) if each set {μ_{alpha} : alpha ∈ Λ} in ba(R) which is pointwise bounded on Δ is norm-bounded in ba (R). If the whole ring R is a Nikodym set, R is said to have property (N), which means that R satisfies the Nikodym-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck’s property (G) and we 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 Nikodym 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 properties (N) and (wN) are equivalents.
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