Theorem - Let $(X, \mathfrak{B}, \mu)$ be a probability space and $T:X \longrightarrow X$ a measure-preserving transformation. The following statements are equivalent:

1. - $T$ is ergodic.
2. - If $f$ is a measurable function and $f\circ T =f$ a.e., then $f$ is constant a.e.
3. - If $f$ is a measurable function and $f\circ T \geq f$ a.e., then $f$ is constant a.e.
4. - If $f \in L^2(X)$ and $f\circ T =f$ a.e., then $f$ is constant a.e..
5. - If $f \in L^p(X)$ , with $p \geq 1$ , and $f\circ T =f$ a.e., then $f$ is constant a.e.

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Let $U_T$ denote the operator induced by $T$ , i.e. the operator defined by $U_T f:= f \circ T$ . The statements above are statements about $U_T$ . The above theorem can be rewritten as follows:

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Theorem - Let $(X, \mathfrak{B}, \mu)$ be a probability space and $T:X \longrightarrow X$ a measure-preserving transformation. The following statements are equivalent:

1. - $T$ is ergodic.
2. - The only fixed points of $U_T$ are the functions that are constant a.e.
3. - If $f$ a measurable function and $U_T f \geq f$ a.e., then $f$ is constant a.e.
4. - The eigenspace of $U_T$ (seen as an operator in $L^p(X)$ , with $p \geq 1$ ) associated with the eigenvalue $1$ , is one-dimensional and consists of functions that are constant a.e.