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

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?