Koopman Operator is Isometry

Theorem
Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.

Let $\map {L^2_\C} \mu$ be the complex-valued $L^2$ space of $\mu$.

Let $U_T : \map {L^2_\C} \mu \to \map {L^2_\C} \mu$ of $T$ be the Koopman operator.

Let $\innerprod \cdot \cdot$ denote the inner product on $\map {L^2_\C} \mu$, i.e.
 * $\ds \forall f, g \in \map {L^2_\C} \mu : \innerprod f g := \int \overline f g \rd \mu$

where $\overline f$ denotes the complex conjugate of $f$.

Then $U_T$ is isometry on the $L^2$ Hilbert space $\struct {\map {L^2_\C} \mu, \innerprod \cdot \cdot}$.

That is:
 * $\ds \forall f, g \in \map {L^2_\C} \mu : \innerprod {U_T f} {U_T g} = \innerprod f g$

Proof
Let $f, g \in \map {L^2_\C} \mu $.

Then: