Strongly nonatomic densities defined by certain matrices

Drewnowski and Paúl proved about ten years ago that for any strongly nonatomic submeasure η on the power set P(ℕ) of the set ℕ of all natural numbers the ideal of all null sets of η has the Nikodym property (NP). They stated the problem whether the converse is true in general. By presenting an example, Alon, Drewnowski and Łuczak proved recently that the answer is negative. Nevertheless, it is of mathematical interest to identify classes of submeasures η such that η is strongly nonatomic if and only if the set of all null sets of η has the Nikodym property. In this context, the authors proved some years ago that this equivalence holds, for instance, if one restricts the attention to the case of densities defined by regular Riesz matrices or by nonnegative regular Hausdorff methods. Also sufficient and necessary conditions in terms of the matrix coefficients are given, that the defined density is strongly nonatomic. In this paper we extend these investigations to the class of generalized Riesz matrices, introduced by Drewnowski, Florencio and Paúl in 1994.