It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See [Parry's Topics in ergodic theory p14](http://farm6.static.flickr.com/5291/5564946645_58634817f9_b.jpg)


Given a probability space $(X,\mathcal{B},\mu)$, can we always find a measure-preserving transformation $T:X \rightarrow X$ such that $\mu$ is $T$-invariant, except the identity?



Given a probability \mu,  can we always find a transformation T s.t. \mu is T-invariant?
For a counterexample to Jon's implicit rewording of your question, consider a purely atomic probability space in which the atoms all have different measure.


For a positive result, consider any purely non atomic probability space. It is measure isomorphic to $[0,1]$ with Lebesgue measure. I guess for non separable purely non atomic probability spaces you just need to apply Maharam's classification theorem. 


