Suppose we want to test $H_0 :\mathbf{X} \sim P_0$ against $H_1 : \mathbf{X} \sim P_1$, where $P_0,P_1$ are two different laws of $\mathbf{X}$.
Let $N$ be the set of points $(\alpha,\beta)$ for which there exists a test $\phi$ such that $E_{P_0}(\phi)=\alpha $ and $E_{P_{1}}(\phi)=\beta$.
Now suppose $P_0 $ and $P_1$ are singular distributions.
Show that $N=[0,1]^2$.
Now, all I could say that as $P_0$ and $P_1$ are singular, the supports of $P_0$ and $P_1$ are disjoint.
This means $\phi(\mathbf{X})=1(f_{0}(\mathbf{x})=0)$ has size $0$ and power $1$.
But what to do next? That is how can I show that $N$ is actually containing all points within the unit square?