On some ‘duality’ of the Nikodym property and the Hahn property

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• 作者：Johann Boos ; Toivo Leiger
• 关键词：Hahn properties ; Nikodym property ; Strongly nonatomic densities ; Densities defined by matrices ; Strong summability by matrices
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2008
• 出版时间：1 May 2008
• 年：2008
• 卷：341
• 期：1
• 页码：235-246
• 全文大小：182 K

文摘

Drewnowski and Paúl proved in [L. Drewnowski, P.J. Paúl, The Nikodým property for ideals of sets defined by matrix summability methods, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94 (2000) 485–503] that for any strongly nonatomic submeasure η on the power set of the ideal has the Nikodym property (NP); in particular, this result applies to densities d_A defined by strongly regular matrices A. Grahame Bennett and the authors stated in [G. Bennett, J. Boos, T. Leiger, Sequences of 0's and 1's, Studia Math. 149 (2002) 75–99] that the strong null domain |A|₀ of any strongly regular matrix A has the Hahn property (HP). Moreover, Stuart and Abraham [C.E. Stuart, P. Abraham, Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems, J. Math. Anal. Appl. 300 (2) (2004) 351–361] 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 ffdcae76b934783169""> and the ideal as well by identifying with the set χ 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 has NP (equivalently, |A|₀ has HP).