Hi everyone!
I am currently studying the basic theory of  measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not be hard, but I am lost... 

> Question: Suppose $T$ is an invertible
> measure-preserving map of standard probability measure
> space $(X,\mu)$. Suppose that $TA=A$
> for all measurable subsets
> $A\subset{}X$, where the equality is
> up to sets of measure $0$. Prove that
> the set of those $x$ where $Tx\neq{}x$
> has measure $0$.