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

Abstract

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 P(N) of N the ideal Z(η)={N∈P(N)|η(N)=0} has the Nikodym property (NP); in particular, this result applies to densities dA 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 dA (defined by A) as submeasure on P(N) and the ideal Z(dA) as well by identifying P(N) 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 Z(dA) has NP (equivalently, |A|0 has HP).