Let $(E,\mathcal E,\mu)$ be a [measure space](/page/Measure%20Space), and let $T:E\to E$ be a measure-preserving map, meaning that $T$ is $\mathcal E/\mathcal E$-measurable and
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\begin{align*}
\mu(T^{-1}(A))=\mu(A)
\end{align*}
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for every $A\in\mathcal E$.
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If $f:E\to\mathbb C$ is $\mathcal E/\mathcal B(\mathbb C)$-measurable and $f\in L^1(E,\mathcal E,\mu)$, then $f\circ T\in L^1(E,\mathcal E,\mu)$ and