Ranks of $\mathcal{F}$-limits of filter sequences

Abstract: We give an exact value of the rank of an $\mathcal{F}$-Fubini sum of filters for the case where $\mathcal{F}$ is a Borel filter of rank $1$. We also consider $\mathcal{F}$-limits of filters $\mathcal{F}<em>i$, which are of the form $\lim</em>\mathcal{F}\mathcal{F}_i=\left{A\subset X: \left{i\in I: A\in\mathcal{F}_i\right}\in\mathcal{F}\right}$. We estimate the ranks of such filters; in particular we prove that they can fall to $1$ for $\mathcal{F}$ as well as for $\mathcal{F}_i$ of arbitrarily large ranks. At the end we prove some facts concerning filters of countable type and their ranks.