I know that if $T(X) = f(W(X))$ for one-to-one $f$, where $W(X)$ is minimal sufficient, then $T(X)$ is also minimal sufficient. But my textbook does not include "almost everywhere" or "almost surely" in this statement, which I think must be a typo?
For example, suppose we have $W(X)$ as being minimal sufficient for $\theta$, and we also have that for some measurable function $f$,
$$P_\theta (T(X) = f(W(X))) = 1$$
Is that enough to conclude $T(X)$ is minimal sufficient? After all, if it only fails in null sets, it doesn't seem to be of importance.