This is proved: Let H be a closed nondiscrete sub- group of an LCA group G, x E G, and Ec Ga a-compact independ- ent subset of G; then Hn(x+G,E) has zero H-Haar measure. This generalizes a result in Rudin, Foiurier analysis on groups; the proof here is quite different from that given by Rudin. THEOREM 1. Suppose that H is a nondiscrete LCA grolp wi hich is a subgroup of a grouip G, x E G and thfat Ec G is an indlependlent set. If (i) G is a LCA gr-oulp such that H w it/h its topology is a closed subgroup in G, and (ii) E is o-compact, then