The authors prove the following theorem: “Let \(\mu\) be a finite positive Borel measure on a metrizable l.c.a. group G which is ergodic with respect to some countable subgroup. If \(\mu\) and its convolution square are not mutually singular, there exists an l.c.a. group H continuously embedded in G such that \(\mu\) is absolutely continuous with respect to Haar measure of H.” For G equal to the torus or the real line the corresponding result is due to G. Brown and W. Moran. In this case H is either discrete or \(H=G,\) and the statement is known as the Generalized Purity Law. The proof given here is more elementary than the one by Brown and Moran.