Limits in compact Abelian groups

摘要

For X a compact Abelian group and B an infinite subset of its dual X, let C_B be the set of all x ∈ X such that < φ(x): φ ∈B > converges to 1. If F is a free filter on X, let D_F = ∪{C_B: B ∈ F}. The sets C_B and D_F are subgroups of X. C_B always has Haar measure 0, while the measure of D_F depends on F. We show that there is a filter F such that D_F has measure 0 but is not contained in any C_B This generalizes previous results for the special case where X is the circle group.