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We classify the pairs of subsets (A,B) of a locally compact abelian groupsatisfying m(A+B)=m(A)+m(B), where m is Haar measure. This generalizes a resultof M. Kneser classifying such pairs under the additional assumption that G iscompact and connected. Our proof combines Kneser's proof with arguments of D.Grynkiewicz, who classified the pairs of subsets (A,B) of abelian groupssatisfying |A+B|=|A|+|B|, where |A| is the cardinality of A.

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