Uniformly Most Powerful Critical Region

Definition

Let $C$ : Critical region of size $\alpha$ for testing the simple hypothesis $H_0$ against an alternative composite hypothesis $H_1$

If $C$ is a best critical region of size $\alpha$ for testing $H_0$ against each simple hypothesis in $H_1$

Then we say $C$ is a uniformly most powerful (UMP) critical region of size $\alpha$