Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of Krieger, we can have three alternatives:
(i) there is a probability measure $\nu\in[\mu]$ in the measure-class $[\mu]$ determined by $\mu$, which is invariant under the action of $T$;
(ii) there is a $\sigma$-finite, non finite measure $\nu\in[\mu]$ in the measure-class $[\mu]$ determined by $\mu$, which is invariant under the action of $T$;
(iii) none of the previous alternatives: namely, the measure-class $[\mu]$ contains no $\sigma$-finite invariant measure (always under the action of the irrational rotation $T$).
In the canonical crossed product construction, the three alternatives above exactly lead to von Neumann factors (because $\mu$ is ergodic) of type $II_1$, $II_\infty$ and $III$, respectively (and the type can be determined only by computing the $ratio\,set$ $r([\mu])$, depending only on the class $[\mu]$).
I'm searching for explicit examples corresponding to (iii) above relative specifically to the irrational rotation $T$.