 Solution

A measure-preserving transformation is a measurable map $T:E\to E$ satisfying $\mu (T^{-1}A)=\mu (A)$ for every measurable $A$. An invariant set satisfies $T^{-1}A=A$ (or equality modulo null sets, in the measure-theoretic convention).

Preimages preserve complements and countable unions. Therefore the whole space is invariant, the complement of an invariant set is invariant, and a countable union of invariant sets is invariant. Thus the invariant sets form a $\sigma$-algebra. The same proof works modulo null sets because a countable union of null exceptional sets is null.