My doubt is the following. Mean ergodic theorem tells us that (there are several versions of the theorem, I will write it down as involving unitary operators on a Hilbert space just because it is simple to me), given a set of Unitary operators $U (x)$, $ x \in \mathbb{R}$ (I assumed continuous index but it could be discrete) on a Hilbert space, the limit
$\lim_{L \to \, + \infty}$ $\frac{1}{2L}$ $\int_{-L}^{L} $dx$ \, $ $U (x)$ $|\psi>$ = $P$ $|\psi>$,
for every $|\psi>$ Hilbert space vector, where $P$ is the projector on the space of vectors invariant under the action of every $U (x)$. Can we say anything instead on
$\lim_{L \to \, + \infty}$ $\frac{1}{2L}$ $\int_{-L}^{L} $dx$ \, $ $f (x)$$U (x)$ $|\psi>$?
where $f (x)$ is a real function. My apologies in advance if it is a trivial question.