I have heard that if one of the distributions is discrete (say Poisson for example) and another one is absolutely continuous (Gaussian as example) it is always possible to construct a statistical criterion, such that it has zero probability of Type $1$ and Type $2$ errors.
Can anybody show how it works?
Let $X$ be a random variable, $H_0: X \sim \text{Poisson}(\lambda)$ and $H_1: X \sim N(\mu,\sigma^2)$. We can define an hypothesis test with the critical region $R=\{x \in \mathbb{R}: x \notin \mathbb{N}\}$. The type I error is the probability of rejection under $H_0$, that is, $P(R|H_0)=0$. This occurs, since under $H_0$, $X$ follows a Poisson distribution and can only be an integer. Also, the type II errors is the probability that you don't reject $H_0$ when it is false, that is, $P(R^c|H_1)=0$. This occurs because, under $H_1$, $X$ has a continuous distribution and $R^c=\mathbb{N}$.
The same type of result occurs whenever there exist two disjoint events, $A_0$ and $A_1$, such that $P(X \in A_0|H_0)=1$ and $P(X \in A_1|H_1)=1$. One can reject $H_0$ whenever $A_1$ occurs and the errors of type I and II will be $0$.