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Strongly nonatomic densities defined by certain matrices. (English) Zbl 1349.46002

Summary: L. Drewnowski and P. J. Paúl [Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94, No. 4, 485–503 (2000; Zbl 1278.46002)] proved about ten years ago that for any strongly nonatomic submeasure \(\eta \) on the power set \(\mathcal {P}(\mathbb {N})\) of the set \(\mathbb {N}\) of all natural numbers the ideal of all null sets of \(\eta \) has the Nikodym property (NP). They stated the problem whether the converse is true in general. By presenting an example, N. Alon et al. [Proc. Am. Math. Soc. 137, No. 2, 467–471 (2009; Zbl 1272.40001)] proved recently that the answer is negative. Nevertheless, it is of mathematical interest to identify classes of submeasures \(\eta \) such that \(\eta \) is strongly nonatomic if and only if the set of all null sets of \(\eta \) has the Nikodym property. In this context, the authors proved some years ago that this equivalence holds, for instance, if one restricts the attention to the case of densities defined by regular Riesz matrices or by nonnegative regular Hausdorff methods. Also sufficient and necessary conditions in terms of the matrix coefficients are given, that the defined density is strongly nonatomic. In this paper we extend these investigations to the class of generalized Riesz matrices, introduced by L. Drewnowski et al. [in: Functional analysis. Proceedings of the first international workshop held at Trier University, Germany, September 26–October 1, 1994. Berlin: de Gruyter. 143–152 (1996; Zbl 0893.46036)].

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
28A10 Real- or complex-valued set functions
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