Invariant-function characterization of ergodicity

A probability measure-preserving system is ergodic exactly when the fixed subspace of its Koopman operator on $L^2$ consists of the almost-everywhere constant functions. Invertibility is unnecessary. If $Uf=f$, invariant level sets of the real and imaginary parts force $f$ to be constant in an ergodic system. Conversely, every invariant set $A$ has $U{1}_A=1_A$, so a constant-only fixed subspace forces $\mu (A)\in \{0,1\}$.

For the noninvertible doubling map on the circle, the ergodicity of integer multiplication on the circle proves this constant-only fixed-space property directly in Fourier coordinates. It illustrates why a unitary inverse is not needed for the characterization.