ON THE NECESSARY AND SUFFICIENT CONDITION OF THE EXISTENCE OF QUASI INVARIANT MEASURES

If E is a separable type-2 Banach space and Esub<0>sub is a linear subspace of E, then the following are equivalent: (a) There exists a probability measure \\mu \] on E, Which is \{E_{\text{0}}}\]-quasi-invariant. (b) There exists a sequence \({X_n}) \subset E\] such that \\sum {{e_n}(\omega ){X_n}} \] converges a.s., where \{{e_n}(\omega )}\] are indepondend identically distributed symmetric stable random variables of index 2,i,e.\E(\exp (it{\kern 1pt} {\kern 1pt} {e_n}(\omega ))) = exp( - \frac{{{t^2}}}{2})\]for all real t, and \{E_{\text{0}}} \subset \{ x,x = \sum {{\lambda _n}{X_n}} ,\forall ({\lambda _n}) \in {l_2}\} \] In this note we prove that \\sum {{\lambda _n}{X_n}} \] is convergent.