‘Duality’ of the Nikodym property and the Hahn property: Densities defined by sequences of matrices

Abstract

The authors investigated in Boos and Leiger (2008) [5] the ‘duality’ of the Nikodym property (NP) of the set of all null sets of the density defined by any nonnegative matrix and the Hahn property (HP) of the strong null domain of it. In this paper, the investigation of the intimated duality is continued by considering densities defined by sequences of nonnegative matrices. These considerations are motivated by the known result that the ideal of the null sets of the uniform density has NP. In this context the general notion of S -convergence of double sequences (cf. Drewnowski, 2002 [8]) containing Pringsheim convergence, Hardy convergence and uniform convergence of double sequences is used.