In many special contexts quasiinvariance of a measure under a one-parameter group of transformations has been established. A remarkable classical general result of A.V. Skorokhod citeSkorokhod74 states that a measure on a Hilbert space is quasiinvariant in a given direction if it has a logarithmic derivative in this direction such that is -integrable for some . In this note we use the techniques of citeSmolyanov-Weizsaecker93 to extend this result to general one-parameter families of measures and moreover we give a complete characterization of all functions for which the integrability of implies quasiinvariance of . If is convex then a necessary and sufficient condition is that is not integrable at .
Cite this article
Michael Scheutzow, Heinrich von Weizsäcker, Which moments of a logarithmic derivative imply quasiinvariance. Doc. Math. 3 (1998), pp. 261–272DOI 10.4171/DM/43