Let G be a nondiscrete
locally compact abelian group, and M(G) the convolution algebra of bounded regular
measures on G. In this paper, the followin g is proved: Let {λk}k=0∞ be a countable
subset of Mc+(G),0≠λ0∈ M0(G), and {Ck}k=0∞ a countable family of σ-compact
subsets of G such that λk(x + Ck) = 0 for all x ∈ G and all k = 0,1,2,⋯ . Then there
exists a nonzero measure σ ∈ M0+(suppλ0) with compact support such that
λk[x + Ck+ Gp(suppσ)] = 0 for all x ∈ G and all k = 0,1,2,⋯ . A consequence of this
result is the following: Let Y be the closed ideal in M(G) which is generated by
∪{L1(λk) : k = 0,1,2,⋯} for some countable subset {λk}λ=0∞ of Mc(G). Then
there exist “fairly many” symmetric maximal ideals in M(G) which contain
∪{L1(μ) : μ ∈ Y }∪ Ma(G) but not M0(G). Here L1(μ) denotes the set
of the measures in M(G) which are absolutely continuous with respect to
|μ|.
