We call two measures equivalent if each is absolutely continuous with respect to the other (cf. [1]). Let G be a locally compact topological group and let μ be a non-negative Baire measure on G (i.e. μ is denned on all Baire sets, finite on compact sets and positive on open sets). We say that μ is stable if μ (E)=0 implies μ(tE)=0 for each t ∈ G. A. M. Macbeath made the conjecture that every stable non-trivial Baire measure is equivalent to the Haar measure. In this paper we prove the following slightly stronger result:

Theorem. Every stable non-trivial measure defined on Baire sets and finite on some open set is equivalent to the Haar measure.