I'm unsure why, given a uniformly most powerful test exists, that the Neyman-Pearson lemma delivers the same critical region for all $\theta_1\in \Omega_1.$ Is it because this is the smallest critical region?
Given a UMP test, why does NP lemma deliver the same critical region for all $\theta_1\in\Omega_1? $
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why ... the same critical region for all theta_1
The test is a single critical region by definition.
It is dependent on the alternative hypothesis $\Omega_1$ (which is a single instance) but independent of whichever value for $\theta_1$ (which can have different values as long as $\theta_1 \in \Omega_1$).
Example: With a one-sided t-test (testing the alternative $\theta > 0$) the rejection criteria is to reject any hypothesis for which the statistic surpasses some critical value $t>t_c$. This critical value/region remains the same no matter what the (unknown) value is for $\theta$.
If a test is uniformly most powerful, then this single region is the best (most powerful) critical region for whatever the value of $\theta_1 \in \Omega_1$. Given some conditions, this region must be unique, and there can not be for some alternative region.
Sextus Empiricus
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