Let $(E,\mathcal E,\mathbb P,T)$ be a probability-preserving system, where $T:E\to E$ is $\mathcal E/\mathcal E$-measurable and measure-preserving. Then $T$ is ergodic if and only if every $\mathcal E/\mathcal B(\mathbb C)$-measurable function $f:E\to\mathbb C$ satisfying
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\begin{align*}
f\circ T=f
\end{align*}
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$\mathbb P$-a.e. is equal to a constant $\mathbb P$-a.e.
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Here $T$ is ergodic means that every set $A\in\mathcal E$ satisfying