Abstract
In this work the necessary and sufficient conditions are given for the quasi-invariance of the distributions of Poisson measures on $X\times\mathbf R^+$ (for arbitrary measurable space~X) with respect to a large group of the scalings of the component~$\mathbf R^+$. It is shown that the class of quasi-invariant measures is far from being exhausted by the measures absolutely continuous with respect to the gamma measure considered in [N. Tsilevich and A. Vershik, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), pp. 163--168] and [N. Tsilevich, A. Vershik, and M. Yor, Prepublication 575, Universites Paris VI & Paris VII, Paris, 2000]. A criterion is given for the absolute continuity of a Poisson measure with respect to another Poisson measure on an arbitrary measurable space.
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