Consider the test problem $$ H_0: \vartheta=\vartheta_0 \quad \text{versus} \quad H_1: \vartheta = \vartheta_1 $$ where $f_{\vartheta_0}: \mathbb{R}^n \rightarrow [0,1] $ and $f_{\vartheta_1}: \mathbb{R}^n \rightarrow [0,1] $ are two probability densities between which we want to choose.
Let $$ \Psi := \left\{ \psi: \mathbb{R}^n \rightarrow [0,1] \; \text{ is a test}\right\} $$ and $$ S := \left\{ \left(\mathbb{E}_{\vartheta_0}\psi, \mathbb{E}_{\vartheta_0}\psi \right): \psi \in \Psi \right\} \subset [0,1]^2 $$
Which points in $S$ correspond with
- tests with significance level $\alpha \in (0, 1)$?
- the uniformly most powerful tests of significance level $\alpha \in (0, 1)$?
Remark: I am a bit lost here because I do not know if for e.g. there is a test for each sensitivity for a given significance.