METRICAL TRANSITIVITY AND NONSEPARABLE EXTENSIONS OF INVARIANT MEASURES

Abstract

Under the Continuum Hypothesis, it is proved that any nonzero $\sigma$-finite metrically transitive invariant measure on a group of cardinality continuum admits a nonseparable invariant extension. An application of this result to the left Haar measure on a $\sigma$-compact locally compact topological group of the same cardinality is also presented.