Given a dynamical system $T:X\to X$, the Koopman operator $U$ is defined on the space of (complex-valued) functions on $X$. So for $f:X\to\mathbb C$, the action of the composition on the function $f$ reads: $U(f):=f\circ T$. Then, since $U$ is then a linear operator in a vector space, one is interested in the eigenfunctions especially those with eigenvalue 1.
But for ergodic transformations $T$, it is a fact that the only $T$-invariant function (i.e. $f\circ T=f$) is an a.e. constant (almost everywhere, up to a zero-measure subset).
So what is the application of this transition to "observable space" for ergodic systems, if there is any? Actually I think, ergodicity is a property mostly assumed for real physical systems, and here the opposite is required?
References: Koopman Operator Spectrum and Data Analysis, Igor Mezic: "Koopman Operator Theory for Dynamical Systems, Control and Data Analytics", Introduction to Koopman operator theory ofdynamical systems, Data-driven spectral decomposition and forecasting of ergodic dynamical systems