Defintions
Let $(X, \mathcal X, \mu, T)$ be a measure preserving system. Let $U_T:L^2_\mu\to L^2_\mu$ be the associated Koopman operator. We will write $\mathcal X_0$ to denote the $\sigma$-algebra of all the $T$-invariant sets in $X$.
We say that $\lambda$ in $\mathbb C$ is an eigenvalue if there is a nonzero $f\in L^2_\mu$ such that $U_Tf= \lambda f$. Note that since $U_T$ is an isometry, we must have $|\lambda| = 1$ whenever $\lambda$ is an eigenvalue. Given an eigenvalue $\lambda$, we say that $f\in L^2_\mu$ is an eigenfunction corresponding to $\lambda$ if $U_Tf=\lambda f$.
It can be checked that the $\sigma$-algebra generated by the set of all the eigenfunctions corresponding to the eigenvalue $1$ is same as $\mathcal X_0$. Let us write $\mathcal X_1$ to denote the $\sigma$-algebra generated by the set of all the eigenfunctions. Clearly, $\mathcal X_0\subseteq \mathcal X_1$.
For each $N\geq 1$ define an operator $A_N:L^2_\mu\to L^2_\mu$ by writing $$A_Nf = \frac{1}{N}\sum_{n=0}^{N-1}U_T^nf$$ Von Neumann ergodic theorem says that $A_N$ converges to $\mathbb E[\cdot|\mathcal X_0]$ in the strong operator topology as $N\to \infty$.
In Lecture 12 of 254A, after Exercise 20, Tao defines a map as follows:
Fix $f\in L^2_\mu$ and define, for each $N\geq 1$, a map $S_{N, f}:L^2_\mu\to L^2_\mu$ by writing $$S_{N, f}g = \frac{1}{N}\sum_{n=0}^{N-1} \langle{g, U_T^nf}\rangle U_T^nf$$ Tao mentions that $S_{N, f}$ converges in the strong topology to an operator $S_f$ with the property that $S_fg$ is an almost periodic function, that is, the closure of $\{U_T^n(S_fg):\ n\in \mathbb Z\}$ is a compact subset of $L^2_\mu$. In Proposition 2 of Lecture 11 Tao shows that if $T$ is $\mu$-ergodic then the almost periodic functions are precisely the $\mathcal X_1$-measurable functions. Thus in the ergodic case $S_fg$ is $\mathcal X_1$-measurable. EDIT: Corollary 3 in Lecture 12 of 254A says that the 'ergodicity hypothesis' in the last line can be dropped.
Question
I do not see the motivation for considering the map $S_{N, f}$. Also, from the above discussion, I feel that there must be some more general averaging kind of map $B_N:L^2_\mu\to L^2_\mu$ which converges in the strong operator topology to $\mathbb E[\cdot|\mathcal X_1]$, much like the von Neumann ergodic theorem. Perhaps $S_{N, f}$ is an avatar of this mythical $B_N$.
Can anybody shed some light. Thanks.