Abstract
Let G be a Hausdorff topological group which is a Baire space. It is proved that if there is a quasi-invariant Radon measure on G then G is locally compact. Examples of non-Baire groups with and without quasi-invariant measures are considered. In particular, it is shown that there is no σ \sigma -finite measure on the Wiener space which preserves sets of measure zero under translation.
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