I am asking whether there is a formulation of quantum ergodicity property of pseudodifferential operators that has the following "form" of Birkhoff's/von Neumann's ergodicity theorems: A sequence of some quantity converges to an integral over the ambient space which depends on the said sequence of quantity. To clarify what I mean, consider the following from Andrew Hassell's Ergodic billiards that are not quantum unique ergodic (second page)
The statement that $\Delta$ is quantum ergodic is the statement that there exists a density-one set $J$ of natural numbers such that the subsequence $(u_j)_{j\in J}$ of eigenfunctions has the following equidistribution property: For each semiclassical pseudodifferential operator $A_h$, properly supported in the interior of $X$, we have $\lim_{j\in J\to\infty}\left<A_{h_j}u_j, u_j\right>_{L^2(X)} = \frac{1}{|S^*X|}\int_{S^*X}\sigma(A)$
where $\sigma(A)$ is the principal symbol of $A$. Then, see one of the classical ergodicity theorems by von Neumann:
Let $(X, F, \mu)$ be a probability space and $T:X\to X$ be a measure preserving transformation. Let $I$ denote the $\sigma$-algebra of $T$-invariant sets. Then for every $f \in L^2(X, F, \mu)$ we have $\frac{1}{n}\sum_{j=0}^{n-1}f\circ T^j\to \mathbb{E}(f\mid I)$ in $L^2$
where $\mathbb{E}(f\mid I) = \int fd\mu$ due to ergodic properties.
Question: Returning back to my initial statement, the $\sigma(A)$ in Hassell's text looks oddly similar to $f$ in the von Neumann's classical ergodicity theorem. Do you happen to know a reference which has formulated QE (or even QUE) results in Birkhoff's/von Neumann's "form"?