Characterization of Ergodicity in terms of Koopman Operator

Theorem
Let $\struct {X, \BB, \mu}$ be a probability space.

Let $T: X \to X$ be a measure-preserving transformation.

Let $\map \MM {X, \R}$ be the set of $\BB$-measurable functions.

Let $\map {\LL^2} \mu$ denote the Lebesgue $2$-space.

Let $U_T : \map \MM {X, \R} \to \map \MM {X, \R}$ be the Koopman operator:
 * $U_T : f \mapsto f \circ T$

Then the following are equivalent:
 * $(1):$ $T$ is ergodic
 * $(2):$ For all $f \in \map \MM {X, \R}$:
 * if $\map {U_T} f = f$, then $f$ is constant $\mu$-a.e.
 * $(3):$ For all $f \in \map \MM {X, \R}$:
 * if $\map {U_T} f = f$ for $\mu$-a.e., then $f$ is constant $\mu$-a.e.
 * $(4):$ For all $f \in \map {\LL ^2} \mu$:
 * if $\map {U_T} f = f$, then $f$ is constant $\mu$-a.e.
 * $(5):$ For all $f \in \map {\LL ^2} \mu$:
 * if $\map {U_T} f = f$ for $\mu$-a.e., then $f$ is constant $\mu$-a.e.

$(1) \implies (3)$
This is clear, since $(3)$ is exactly the definition of ergodicity.

$(3) \implies (2)$ and $(5) \implies (4)$
These are a direct consequence of.

If $\map {U_T} f = f$, the same holds especially $\mu$-almost everywhere, since:

$(2) \implies (4)$ and $(3) \implies (5)$
In view of :
 * $\map {\LL ^2} \mu \subseteq \map \MM {X, \R}$

The claims follow from this.

$(4) \implies (1)$
Let $A \in \BB$ be such that $T^{-1} \sqbrk A = A$.

Let $\chi_A : X \to \set {0, 1}$ be the characteristic function of $A$.

Note that $\chi_A^2 = \chi_A$, as $0^2=0$ and $1^2=1$.

In particular, $\chi_A \in \map {\LL^2} \mu$, since:
 * $\ds \int \chi_A^2 \rd \mu = \map \mu A < + \infty$

Moreover:


 * $\chi_A \circ T = \chi_A$

since for all $x \in X$:

That is, $\map {U_T} f = f$ by.

Therefore, $\chi_A$ is constant $\mu$-almost everywhere.

The claim follows from this, since by :
 * $A = \set {x \in X : \map {\chi_A} x = 1}$

and:
 * $X \setminus A = \set {x \in X : \map {\chi_A} x = 0}$