Optimal unbiased estimators in additive models with bounded errors are deterministic

In an additive model $X=\vartheta+\varepsilon$, $\vartheta\in\Theta\subset{\mathbf R}^k$, let the errors $\varepsilon$ have a compactly supported but otherwise arbitrary known joint distribution. Let $g$ be a uniformly minimum variance unbiased estimator for its own expectation $\gamma(\vartheta)$. We show that under mild regularity conditions, $g$ is deterministic: for every $\vartheta\in\Theta$, $g(X)=\gamma(\vartheta)$ almost surely. Our proof uses a lemma on entire quotients of Fourier transforms which might be of independent interest.