The Lehmann–Scheffé theorem from Wikipedia
The theorem states that any estimator which is unbiased for a given unknown quantity and which is based on only a complete, sufficient statistic (and on no other data-derived values) is the unique best unbiased estimator of that quantity. ...
Formally, if $T$ is a complete sufficient statistic for $θ$ and $E(g(T)) = τ(θ)$ then $g(T)$ is the minimum-variance unbiased estimator (MVUE) of $τ(θ)$.
I was wondering why the last sentence is correct?
My question boils down to:
If $T$ is a complete sufficient statistic (unbiased?) for $θ$, is $g(T))$ a complete sufficient unbiased estimator of $E(g(T)) = τ(θ)$?
Thanks and regards!