This is example in Hogg and Craig's book page 254
I have shown that the best critical region is $\sum X_i>=3$.
is equivalent to $\sum X_i>=3$, and this is the critical region for testing for each simple hypothesis in $H_1$ and therefore from here can't I say that this is a uniformly most powerful test.
Have they done it like this because since Poisson distribution is discrete we can't find critical value so that Pr(type 1 error) is exactly equal to significance level. But in this case isn't the critical value selected so that size=largest possible values less than significance level. So in that case shouldn't the best critical region be, $\sum X_i>=4$