Quasi-invariant measures for topological dynamical systems

It is proved that for a topological dynamical system to admit an ergodic quasi-invariant measure of type III (a measure which is not equivalent to any $\sigma$-finite invariant measure) it is necessary and sufficient that this system have a recurrent point. For systems with a recurrent point, it is shown that there exist a nondenumerable number of pairwise singular ergodic quasi-invariant measures of type III.