I have seen the equality $E[Y \mid f(W)] = E[Y \mid W]$ being used in many econometric derivations. For example, in demonstrating the control function approach for identification (here is the Wikipedia article) $E[U \mid X,V] = E[U \mid \pi(Z) + V, V] = E[U \mid \ Z, V]$.
If generally $\sigma(f(W)) \subseteq \sigma(W)$, how can $E[Y \mid \sigma(f(W))] = E[Y \mid \sigma(W)]$ hold?
If $f \equiv 0$ then $E(Y|f(W))=EY$ so the equality does not hold always. It holds if $\sigma (f(W))=\sigma (W)$ which is true if $f$ is bijective and $f^{-1}$ is also Borel measurable