Limits in compact abelian groups

2004/08/09 by Joan E. Hart, Hart, Joan E., Kenneth Kunen +1
Mathematics · #22C05 #54H11 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #math.GN #math.GR #msc:22C05 #msc:54H11

paper · pdf · doi:10.48550/arxiv.math/0408115

14 pages

arxiv created 2004/08/09 · arxiv updated 2009/12/01

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 <phi(x) : phi ∈ B> 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.

Related

• Limits in Function Spaces and Compact Groups
• On Wiener's lemma for locally compact abelian groups
• Sequences and filters of characters characterizing subgroups of compact abelian groups
• Countably compact group topologies on arbitrarily large free Abelian groups
• Locally compact abelian groups admitting non-trivial quasi-convex null sequences