On the Uniqueness of Non-locally Finite Invariant Measures on a Topological Group

An example proves that the Uniqueness theorem for non-locally finite invariant measures on a topological group does not hold, In this paper a uniqueness theorem for locally σ-finite invariant measures is given and it is proved that two invariant measures with the same sets of finite measure on a non-compact group are mutually absolutely continuous. This result is improved in the case of Hausdorff measures.