On the structure of measurable filters on a countable set

نویسنده

  • Tomek Bartoszynski
چکیده

A combinatorial characterization of measurable filters on a countable set is found. We apply it to the problem of measurability of the intersection of nonmeasurable filters. The goal of this paper is to characterize measurable filters on the set of natural numbers. In section 1 we introduce basic notions, in section 2 we find a combinatorial characterization of measurable filters, in section 3 we study intersections of filters and finally section 4 is devoted to filters which are both null and meager. Through this paper we use standard notation. ω denotes the set of natural numbers. For k, n ∈ ω let [n, k] = {i < ω : n ≤ i ≤ k}. For n ∈ ω, 2 (2) denotes the set of 0-1 sequences of length n(ω), also let 2 = ⋃ n∈ω 2 . For any sequences s, t ∈ 2 let st denote their concatenation. For s ∈ 2 let [s] = {x ∈ 2 : s ⊂ x}. The family {[s] : s ∈ 2} is a base of the space 2. We will often identify a set [s] with a sequence s and we will also identify subsets of ω with their characteristic functions. Filters considered in this paper are assumed to be nonprincipal. We identify filters on ω with sets of characteristic functions of its elements. In this way the question about measurability makes sense. Finally let quantifiers “∃” and “∀” denote “for infinitely many” and “for all except finitely many” respectively.

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تاریخ انتشار 1999