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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:

[1] Bennett, G.; Boos, J.; Leiger, T., Sequences of 0’s and 1’s, Studia Math., 149, 75-99 (2002) · Zbl 0995.46010
[2] Boos, J., Classical and Modern Methods in Summability (2000), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 0954.40001
[3] Boos, J.; Parameswaran, M. R., The potency of weighted means: Addendum to a paper of Kuttner and Parameswaran, J. Anal., 7, 219-224 (1999) · Zbl 0957.40003
[4] Boos, J.; Zeltser, M., Sequences of 0’s and 1’s. Classes of concrete ‘big’ Hahn spaces, Z. Anal. Anwendungen, 22, 4, 819-842 (2003) · Zbl 1061.46004
[5] Drewnowski, L.; Paúl, P. J., The Nikodým property for ideals of sets defined by matrix summability methods, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.), 94, 485-503 (2000) · Zbl 1278.46002
[6] Hill, J. D.; Sledd, W. T., Approximation in bounded summability fields, Canad. J. Math., 20, 410-415 (1968) · Zbl 0162.08001
[7] Kuttner, B.; Parameswaran, M. R., A class of conservative summability methods that are not potent, J. Anal., 1, 91-98 (1993) · Zbl 0779.40004
[8] Kuttner, B.; Parameswaran, M. R., Potent conservative summability methods, Bull. London Math. Soc., 26, 297-302 (1994) · Zbl 0812.40003
[9] Kuttner, B.; Parameswaran, M. R., A class of weighted means as potent conservative methods, J. Anal., 4, 161-172 (1996) · Zbl 0880.40003
[10] Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Math., 80, 167-190 (1948) · Zbl 0031.29501
[11] Meyer-König, W., Untersuchungen über einige verwandte Limitierungsverfahren, Math. Z., 52, 257-304 (1949) · Zbl 0041.18403
[12] Sember, J., Families of sequences of 0s and 1s in FK-spaces, Canad. Math. Bull., 33, 18-23 (1990) · Zbl 0725.46004
[13] Stuart, C. E.; Abraham, P., Generalizations of the Nikodym boundedness and Vitali-Hahn-Saks theorems, J. Math. Anal. Appl., 300, 2, 351-361 (2004) · Zbl 1081.28005
[14] Wilansky, A., Modern Methods in Topological Vector Spaces (1978), McGraw-Hill International Book Co.: McGraw-Hill International Book Co. New York · Zbl 0395.46001
[15] Wilansky, A., Summability through Functional Analysis, Notas Mat., vol. 85 (1984), North-Holland: North-Holland Amsterdam · Zbl 0531.40008
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