In this paper, following the methods of Connor [connor], we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely [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 [jc]. We mainly investigate...

We consider the Katětov order between ideals of subsets of natural numbers ("${\le }_K$") and its stronger variant-containing an isomorphic ideal ("⊑ "). In particular, we are interested in ideals for which ${\le }_K\Rightarrow ⊑$ for every ideal . We find examples of ideals with this property and show how this property can be used to reformulate some problems known from the literature in terms of the Katětov order instead of the order "⊑ " (and vice versa).