We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called μ-statistical convergence/Cauchy condition and convergence/Cauchy condition in μ-density, studied for real numbers in our recent paper ) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz  and Mursaleen and Edely . We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case  but at the same time, like , shows that symmetry is not essential for any result of  and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).
Key words and phrases
asymmetric metric spaceapproximate metric axiom (AMA)double sequencesforward and backward μ-statistical convergence/divergence/Cauchy conditionconvergence/divergence/Cauchy condition in μ-densitycondition (APO2)