Neyman-Pearson lemma for discrete distributions

The proof of the Neyman-Pearson lemma relies on finding some threshold $\kappa$ such that $P(\Lambda (X)\ge \kappa )=\alpha$ where $\Lambda$ is the likelihood ratio. If $P$ is continuous, then this is possible. But what if the distribution is discrete? In that case, we use randomization.

We still use a variant of the likelihood-ratio test, but it has the following form:

$$\varphi (x)=\{\begin{array}{ll}1,& \Lambda (x)>\kappa ,\\ \gamma ,& \Lambda (x)=\kappa ,\\ 0,& \Lambda (x)<\kappa ,\end{array}$$

for some threshold $\kappa$ and where $\gamma$ is chosen such that $\varphi$ has type-I error exactly $\alpha$. Therefore, $\gamma$ solves ${\mathbb{E}}_Q[\varphi (X)]={\sum }_{x:\Lambda (x)>\kappa }q(x)+{\sum }_{x:\Lambda (x)=\kappa }\gamma =\alpha$. This might be considered a form of external randomization.