# If a statistic can be written as a function of a minimal sufficient statistic almost everywhere, is it minimal sufficient?

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.

• This is close to nit-picking: the random variable $X$ is defined (as a function) almost everywhere rather than everywhere so the answer is yes. – Xi'an Oct 28 '18 at 16:19