ADMISSIBLE TRANSLATES OF MEASURES ON A TOPOLOGICAL GROUP

Abstract

The algebraical and the measure theoretical properties of admissible (singular) translates on a topological group are studied. It is shown the sets of all admissible (singular) translates such as RA(μ), LA(μ), TA(μ), RE(μ), LE(μ), TE(μ), RS(μ), LS(μ), TS(μ), are Borel subsets of the topological group. If μ₁ is right quasi-invariant and μ₂ is left quasi-invariant then μ₁, μ₂ are equivalent. In a locally compact case each right (or left) quasi-invariant measure is equivalent to the Haar measure. For a right quasi-invariant measure μ on a separable group, a right translation invariant measure μ₀ which is equivalent to μ is constructed. Using this fact the following result is proved, which gives some informations about the measure theoretical size of the set of admissible translates.
THEOREM 1. For each Radon probability measure μ on a separable group G, μ(RE(μ))=0, or RE(μ) is a locally compact group and the restriction μ | RE(μ) is equivalent to the Haar measure on RE(μ).