Two valued measures and summability
In this note we introduce deflnltlons or u-statist1cal and u-density convergence where n is a {0, l)-valued finitely additive measure defined on a field of subsets of N. It is shown that these two surnrnability methods are equivalent iff the measure which generale them is "nearly" countably additive which occurs iff the ideal associated wU.h the measure contalns no proper dense subideal which contalns CQ. It Is also shown that the methods of strong, Statistical and density convergence with respect to a nonnegative regulär matrix summability method are equivalent for bounded sequences. AMS Subject Classification: 40D25, 40025, 40A05 This paper Is a continuation of l4l, where it is observed that Statistical convergence, convergence in density and strong summability with respect to the Cesaro matrix are equivalent on the bounded sequences. In this note we show that the Cesaro matrix can be replaced by any nonnegative regulär summability matrix as well as expose and dlscuss a structure common to the notlons of Statistical convergence, convergence in density and strong summabllty for nonnegative regul»^ summability methods. The common thread throughout the dlscussion Is an analysis of the Ideals generated by these methods and of the {0,1)-valued finitely additive measures associated with these Ideals. We also Show how to establish the Statistical convergence and convergence in density with respect to an arbitrary proper ideal of FIO» that contalns CQ and describe the closure of such an Ideal in ê .̂