In ergodic theory, why does the defintion of an ergodic transformation $T$, why do I have to claim that it is measure-preserving? E.g.
$T$ is ergodic if $\mathbb{P}(A) \in \{0,1\}$ for all $A$ with $T^{-1} (A) = A$
Couldn't I also have this definition without $T$ being measure-preserving ($\mathbb{P} \circ T = \mathbb{P}$) ?