Limits in function spaces and compact groups

For B an infinite subset of ω and X a topological group, let CBX be the set of all x∈X such that 〈xn:n∈B〉 converges to 1. CBT always has measure 0 in the circle group T. If F is a filter of infinite sets, let DFX=⋃{CBX:B∈F}. Then CBX and DFX are subgroups of X when X is Abelian. We show that there is a filter F such that DFT has measure 0 but is not contained in any CBT. In contrast, for any compact metric group X, there is a filter G such that DGX=X; this follows from a more general result in this paper on limits in function spaces. Also, we show that some of the properties of DFX, for arbitrary compact groups X, are determined by the special cases X=T or X=Tω.