Abstract. It is shown that the uniqueness property of probability invariant measures is preserved under the operation of Cartesian products. A simi-lar question is investigated for inductive limits of nonzero σ-finite invariant measures. 2000 Mathematics Subject Classification: 28A05, 28D05. Key words and phrases: Invariant measure, metrically transitive measure, uniqueness property, product of measures, inductive limit of measures. It is well known that the uniqueness property for invariant measures plays a significant role in various questions of modern analysis and general topology. For instance, the Haar measure on a locally compact topological group has the uniqueness property and this fact gives rise to many important consequences in abstract harmonic analysis, in the theory of dynamical systems, etc. (see, e.g., [1], [2]). The main purpose of this article is to consider the uniqueness property of in-variant measures from the general point of view and to investigate this property