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Let G be a topological group of second category and having cardinality at most that of the continuum. Let B be some a-algebra of subsets of G such that (G, B) is a measurable group. For a probability measure P on (G, B), write P for the measure defined by Pg(E) = P (gE), E e B. The aim of this paper is to prove the following: if on (G, B) there exists an inner-regular probability measure P such that Pg m >i Pg for every g E G on the a-algebra S n B.

On a characterization of locally compact groups of second category, assuming the continuum hypothesis