An inverse theorem: when the measure of the sumset is the sum of the\n measures in a locally compact abelian group

We classify the pairs of subsets (A,B) of a locally compact abelian group\nsatisfying m(A+B)=m(A)+m(B), where m is Haar measure. This generalizes a result\nof M. Kneser classifying such pairs under the additional assumption that G is\ncompact and connected. Our proof combines Kneser's proof with arguments of D.\nGrynkiewicz, who classified the pairs of subsets (A,B) of abelian groups\nsatisfying |A+B|=|A|+|B|, where |A| is the cardinality of A.\n