 Solution

A measurable set is invariant when ${\theta }^{-1}A=A$, or modulo a null set in the measure-theoretic version. A measurable function is invariant when $f\circ \theta =f$, similarly almost everywhere if appropriate. The map is a measure-preserving transformation when $\mu ({\theta }^{-1}A)=\mu (A)$ for every measurable set. A measure-preserving map is ergodic when every invariant measurable set has measure zero or has null complement. On a finite positive measure space, normalize the measure to obtain the equivalent probability-space definition.