On sets not belonging to algebras

Let script A sign₁, . . ., script A sign_n script A sign_n+1 be a finite sequence of algebras of sets given on a set X, ∪ⁿ_k=1 script A sign_k ≠ B-frakfur sign (X), with more than 4/3n pairwise disjoint sets not belonging to script A sign_n+1. It was shown in [4] and [5] that in this case ∪_k-1ⁿ⁺¹ script A sign_k ≠ B-frakfur sign(X). Let us consider, instead script A sign_n+1, a finite sequence of algebras script A sign_n+1, . . . , script A sign_n+l. It turns out that if for each natural i ≤ l there exist no less than 4/3(n + l) - l/24 pairwise disjoint sets not belonging to script A sign_n+i, then ∪_k=1ⁿ⁺¹ script A sign_k ≠ B-frakfur sign(X). But if l ≥ 195 and if for each natural i ≤ l there exist no less than 4/3(n + l) - l/15 pairwise disjoint sets not belonging to script A sign_n+i, then ∪_k=1^n+lscript A sign _k ≠ B-frakfur sign(X). After consideration of finite sequences of algebras, it is natural to consider countable sequences of algebras. We obtained two essentially important theorems on a countable sequence of almost σ-algebras (the concept of almost σ -algebra was introduced in [4]).