It is shown, that if G is a LCA group and if H is a nondiscrete LCA group then there exists a proper closed subalgebra of the measure algebra of H (independent of the choice of G) in which the range of every homomorphism of the group algebra of G into the measure algebra of H is contained. Throughout this paper, G and H denote LCA groups and G and H denote their dual groups, respectively. X(H) is the set of all the locally compact group topologies of H which are at least as strong as the original one of H. For each % e X(H), if we denote by H7 a LCA group with underlying group H and topology t, the natural continuous isomorphism of HT onto H, x e ?Ti—>;t £ H, induces a natural norm-preserving im- bedding of L1(HT) into M(H), which we also denote by Lx(Hr). For the other notations and terminologies which we need in this paper, we follow (6). The author would like to express his thanks to the referee. His kind advice enabled the author to make this paper more readable. Theorem. If h is a homomorphism of Ll(G) into M(H), then there exist finitely many elements tx, t2, • ■ • , rn e X(H) such that the range of h is contained in 2?=i Lx(Hr'). For the proof of the theorem, we essentially use Cohen's results, which determine all the homomorphisms of L1(G) into M(H) by the notion of the coset ring and piecewise affine maps (cf. (1), (2), (3) and (6, Chapters 3 and 4)). If A is a homomorphism of L1(G) into M(H), Cohen's theorem asserts that there exist Y, an element of the coset ring of H, and a piecewise affine map a from Y into G such that