Existence of maximin tests.

5 Let (X , A) be a Euclidean sample space, and let the distributions Pθ , θ ∈ , be dominated by a σ-finite measure over (X , A).

For any mutually exclusive subsets H , K of  there exists a level-α test maximizing (8.2).

[Let β = sup[infk Eθϕ(X)], where the supremum is taken over all level-α tests of H : θ ∈ H . Let ϕn be a sequence of level-α tests such that infK Eθϕn(X) tends to β. If ϕni is a subsequence and ϕ a test (guaranteed by Theorem A.5.1 of the Appendix) such that Eθϕni(X) tends to Eθϕ(X) for all θ ∈ , then ϕ is a level-α

test and infk Eθϕ(X) = β.]