Ranks of -limits of filter sequences

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