Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 26J / a

A probability measure $\mu$ is $T$-invariant whenEquivalently, the pushforward measure satisfies $T_{\ast }\mu =\mu$, so $T$ is a measure-preserving transformation.

Assuming $\mu$ is $T$-invariant, $T$ is an ergodic transformation when every measurable set invariant modulo null sets is trivial:Equivalently, the invariant sigma-algebra is trivial modulo null sets.