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Limits in compact abelian groups

Joan E. Hart·Kenneth Kunen
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Abstract

Let X be compact abelian group and G its dual (a discrete group). If B is an infinite subset of G, let C_B be the set of all x in X such that converges to 1. If F is a free filter on G, let D_F be the union of all the C_B for B in 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.

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Joan E. Hart, Kenneth Kunen. 2004-08-09. Limits in compact abelian groups. https://arxiv.org/abs/math/0408115

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