Two valued measure and summability of double sequences

summary:In this paper, following the methods of Connor \cite {connor}, we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely \cite {moe}) to $\mu $-statistical convergence and convergence in $\mu $-density using a two valued measure $\mu $. We also apply the same methods to extend the ideas of divergence and Cauchy criteria for double sequences. We then introduce a property of the measure $\mu $ called the (APO$_2$) condition, inspired by the (APO) condition of Connor \cite {jc}. We mainly investigate the interrelationships between the two types of convergence, divergence and Cauchy criteria and ultimately show that they become equivalent if and only if the measure $\mu $ has the condition (APO$_2$)