Abstract. We investigate the extent to which sequence spaces are determined by the sequences of 0's and 1's that they contain.

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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 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 P(N) and the ideal Z(d A ) 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(d A ) has NP (equivalently, |A| 0 has HP).

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