Let G be a locally compact abelian group with a Haar measure Xq . A function f on G is said to be mean-periodic if there exists a nonzero finite regular measure p. of compact support on G such that f * p = 0 . It is known that there exist no nontrivial integrable mean periodic functions on R" . We show that there exist nontrivial integrable mean periodic functions on G provided G has nontrivial proper compact subgroups. Let f 6 LX(G) be mean periodic with respect to a nonzero finite measure p

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... compact support. If p.(G) ^ 0 and Ac(supp(/z)) > 0 , then there exists a compact subgroup K of G such that f*X/c = 0 , i.e., f is mean periodic with respect to Xk , where Xk denotes the normalized Haar measure of K . When G is compact, abelian and meterizable, we show that there exists continuous (hence integrable and almost periodic) functions on G that are not mean periodic.