## 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

### Citations:

Zbl 0995.46010; Zbl 1081.28005
Full Text:

### References:

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