We classify the pairs of subsets
A
A
,
B
B
of a locally compact abelian group
G
G
satisfying
m
∗
(
A
+
B
)
=
m
(
A
)
+
m
(
B
)
m_*(A+B)=m(A)+m(B)
, where
m
m
is the Haar measure for
G
G
and
m
∗
m_*
is inner Haar measure. This generalizes M. Kneser’s classification of such pairs when
G
G
is assumed to be connected. Recently, D. Grynkiewicz classified the pairs of sets
A
A
,
B
B
satisfying
|
A
+
B
|
=
|
A
|
+
|
B
|
|A+B|=|A|+|B|
in an abelian group, and our result is complementary to that classification. Our proofs combine arguments of Kneser and Grynkiewicz.