If the sample space X is Euclidean and P0, P1 have densities with respect to Lebesgue measure, there exists a nonrandomized most powerful test for testing P0 against P1 at every significance level α.

13 [This is a consequence of Theorem 3.2.1 and the following lemma.14 Let f ≥ 0 and A f (x) dx =

a. Given any 0 ≤ b ≤

a, there exists a subset B of A such that B f (x) dx = b.]