On quasi-invariant measures in topological vector spaces

Abstract

We give a complete solution of the fundamental problem of existence of quasi-invariant measures in innite dimensional vector spaces considered and partially solved by I.M. Gelfand in 1967 [4]. Our general result shows particularly that in all innite dimensional usual vector spaces, in particular,in all Fr echet, Banach or Hilbert spaces and in all spaces of distributions, the only quasi-invariant measure is null measure.