ergodicity of a map in terms of its induced operator
Theorem  Let $(X,\U0001d505,\mu )$ be a probability space and $T:X\u27f6X$ a measurepreserving 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.
$$
Let ${U}_{T}$ denote the operator induced by $T$ (http://planetmath.org/OperatorInducedByAMeasurePreservingMap), i.e. the operator defined by ${U}_{T}f:=f\circ T$. The statements above are statements about ${U}_{T}$. The above theorem can be rewritten as follows:
$$
Theorem  Let $(X,\U0001d505,\mu )$ be a probability space and $T:X\u27f6X$ a measurepreserving 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 onedimensional and consists of functions that are constant a.e.
Title  ergodicity of a map in terms of its induced operator 

Canonical name  ErgodicityOfAMapInTermsOfItsInducedOperator 
Date of creation  20130322 17:59:22 
Last modified on  20130322 17:59:22 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  6 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 47A35 
Classification  msc 37A30 
Classification  msc 37A25 
Classification  msc 28D05 