ergodicity of a map in terms of its induced operator

Theorem - Let $(X,𝔅,\mu )$ be a probability space and $T:X⟶X$ 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. (http://planetmath.org/AlmostSurely), then $f$ is constant a.e.

3. - If $f$ is a measurable function and $f\circ T\ge f$ 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 $p\ge 1$, 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 $T$ (http://planetmath.org/OperatorInducedByAMeasurePreservingMap), 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:

$$

Theorem - Let $(X,𝔅,\mu )$ be a probability space and $T:X⟶X$ 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 $U_{T}f\ge f$ a.e., then $f$ is constant a.e.

4. - The eigenspace of $U_T$ (seen as an operator in $L^p(X)$, with $p\ge 1$) associated with the eigenvalue $1$, is one-dimensional and consists of functions that are constant a.e.