= 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 \b[the invariant sets form a $\sigma$-algebra]. The same proof works modulo null sets because a countable union of null exceptional sets is null.
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