 Solution

For a rejection function $\phi (x)\in [0,1]$, its power is $\beta (\theta )={\mathbb{E}}_{\theta }\phi (X)$. A test has level at most $\alpha$ if ${\sup }_{\theta \in H_0}\beta (\theta )\le \alpha$; its size is that supremum. In a simple null the significance probability is just the rejection probability under that null.

For two simple hypotheses with joint sample densities $f_0,f_1$, the Neyman–Pearson lemma states that a test rejecting where $f_1>k{f}_0$, accepting where $f_1<k{f}_0$, and randomizing if necessary on equality to have size $\alpha$, is most powerful among level-$\alpha$ tests. Let ${\phi }_{\ast }$ be that test and $\phi$ any competitor. Pointwise,Integration givessince $k\ge 0$, the first test has size $\alpha$, and the other has size at most $\alpha$. This proves the claimed power bound, allowing null-density-zero regions as well. Equality implies agreement away from the likelihood-ratio boundary, up to null sets, when the inequalities are strict.