Let $\Omega = \mathbb{R}^n$ and probability measures $\mathbb{P}\small{1}, \mathbb{P}\small{2}$ are absolutely continuous with regard to the Lebesgue measure.
Does a non-randomized most powerful test exist if we test a hypothesis $H\small1 = \{\mathbb{P} = \mathbb{P}\small{1}\}$ with an alternative hypothesis $H\small2 = \{\mathbb{P} = \mathbb{P}\small{2}\}$ for every significance level $\alpha$ ?