I’m currently reading about the Neyman-Pearson Lemma for the MP level-$\alpha$ test. My question is, when does there not exist a MP level-$\alpha$ test?
For example, suppose that $X_1,...,X_n$ are iid $N(\mu,\sigma^2)$ where $\sigma$ is known, $\mu_0<\mu_1$, and $\mu_0=\mu$ is the null hypothesis. The particular question I’m working on asks if there’s any $\alpha$ for which a MP test exists, but after setting up the ratio of likelihood functions with alternative over null, I’m not sure why such a test wouldn’t exist. What must I check for and why?