# Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant?

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?

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The identity map. –  Jon Bannon Mar 27 '11 at 20:07
You are right, I should exclude the identity. –  Choi Mar 27 '11 at 20:12

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.