Let $(E,\mathcal E,\mu)$ be a [measure space](/page/Measure%20Space), and let $T:(E,\mathcal E)\to(E,\mathcal E)$ be a measurable map satisfying $\mu(T^{-1}(A))=\mu(A)$ for every $A\in\mathcal E$. Define $T^0:=\operatorname{id}_E$, and define the iterates recursively by $T^{k+1}:=T^k\circ T$ for every $k\in\mathbb N\cup\{0\}$. Then, for every $k\in\mathbb N\cup\{0\}$, the map $T^k:(E,\mathcal E)\to(E,\mathcal E)$ is measurable and satisfies $\mu((T^k)^{-1}(A))=\mu(A)$ for every $A\in\mathcal E$.