Rotationally-quasi-invariant measures on the dual of a Hilbert space (Q1070638)

Let H be real Hilbert space, \(H^ a\) the algebraic dual space of H, \({\mathcal B}\) the \(\sigma\)-field generated by cylinder sets of \(H^ a\), O(H) the group of all orthogonal operators on H, and \({}^ tU\) the algebraic transpose of \(U\in O(H)\). For \(U\in O(H)\) and a probability measure \(\mu\) on \({\mathcal B}\), the measure \(\mu_ U\) is defined as \(\mu_ U(B)=\mu (^ tU(B)),\) \(B\in {\mathcal B}.\) In the paper it is proved, that for any rotationally-quasi-invariant measure \(\mu\) (for all \(U\in O(H)\), \(\mu_ U\) and \(\mu\) are absolutely continuous to each other) there exists a rotationally-invariant measure \(\nu\) \((\nu_ U=\nu\) for all \(U\in O(H))\) which is equivalent with \(\mu\).