Remarks on the imprimitivity theorem for nonlocally compact Polish groups

In this paper we analyze the possibility of establishing a Theorem of Imprimitivity in the case of nonlocally compact Polish groups. We prove that systems of imprimitivity for a Polish group G based on a locally compact homogeneous G-space M ≡ G/H equipped with a quasi-invariant probability measure μ, are in one-to-one correspondence with elements of the space of the first cohomology of the group G of equivalence classes of continuous cocycles. As a corollary, we have the complete Imprimitivity Theorem in the case of discrete countable homogeneous G-spaces equipped with a quasi-invariant measure. Finally, we outline the possibility of establishing the complete Imprimitivity Theorem for particular classes of Polish groups. These examples cover the case of (separable) Fréchet spaces, for which it is shown that the complete Imprimitivity Theorem holds as well.