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On some ‘duality’ of the Nikodym property and the Hahn property. (English) Zbl 1138.28001

Summary: L. Drewnowski and P. J. Paúl [The Nikodým property for ideals of sets defined by matrix summability methods, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94, 485–503 (2000)] proved that for any strongly nonatomic submeasure \(\eta \) on the power set \(\mathcal P(\mathbb N)\) of \(\mathbb N\) the ideal \(\mathcal Z(\eta) = \{N\in \mathcal P(\mathbb N)| \eta(N) = 0\}\) has the Nikodym property (NP); in particular, this result applies to densities \(d_A\) defined by strongly regular matrices \(A\). G. Bennett and the authors stated [Stud. Math. 149, No. 1, 75–99 (2002; Zbl 0995.46010)] that the strong null domain \(|A|_{0}\) of any strongly regular matrix \(A\) has the Hahn property (HP). Moreover, C. E. Stuart and P. Abraham [J. Math. Anal. Appl. 300, No. 2, 351–361 (2004; Zbl 1081.28005)] pointed out that the said results are in some sense dual and that the last one follows from the first one by considering the density \(d_A\) (defined by \(A\)) as submeasure on \(\mathcal P(\mathbb N)\) and the ideal \(\mathcal Z(d_A)\) as well by identifying \(\mathcal P(\mathbb N)\) with the set \(\chi \) of sequences of 0’s and 1’s. In this paper we aim at a better understanding of the intimated duality and at a characterization of those members of special classes of matrices \(A\) such that \(\mathcal Z(d_A)\) has NP (equivalently, \(|A|_{0}\) has HP).

MSC:

28A33 Spaces of measures, convergence of measures
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
46A45 Sequence spaces (including Köthe sequence spaces)
40C05 Matrix methods for summability
28B05 Vector-valued set functions, measures and integrals
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References:

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