Logarithmic density and measures on semigroups

Davenport and Erdos [3] proved that every set A of integers with the property that a is an element of A implies an is an element of A for all n (multiplicative ideal) has a logarithmic density. I generalized [5] this result to sets with the property that if for some numbers a, b, n we have a is an element of A, b is an element of A and an is an element of A, then necessarily bn is an element of A, which I call quasi-ideals. Here a new proof of this theorem is given, applying a result on convolution of measures on discretes semigroups. This leads to further generalizations, including an improvement of a result of Warlimont [8] on ideals in abstract arithmetic semigroups.