In this dissertation we investigate the following problem in ergodic theory: Let X be a Polish space and T:X→X a homeomorphism. We ask whether there exists a non-atomic Borel probability measure m on X, ergodic and quasi-invariant under T, such that the cohomology class of some preassigned continuous function f:X→S1 is trivial. The cohomology class of f is trivial and f is called a coboundary of T, if the equation hTx=f x˙hx m-almosteverywhere admits a measurable solution h:X→S1 In the presence of a non-periodic recurrent point x0∈X , we establish the existence of such a measure m . Indeed, our approach, which is based on a construction of Katznelson and Weiss, yields a continuum of such measures all of which are of type IIinfinity and pointwise orthogonal. And the same statement is true for measures of type III