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 \mu on a Hilbert space is quasiinvariant in a given direction if it has a logarithmic derivative \beta in this direction such that e^{a|\beta|} is \mu -integrable for some a > 0 . 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 \psi:[0,\infty) \rightarrow [0,\infty) for which the integrability of \psi(|\beta|) implies quasiinvariance of \mu . If \psi is convex then a necessary and sufficient condition is that \log \psi(x)/{x^2} is not integrable at \infty .