Existence of maximin tests.

4 Let (X , A) be a Euclidean sample space, and let the distributions Pθ, θ ∈ Ω, be dominated by a σ-finite measure over (X , A). For any mutually exclusive subsets ΩH, ΩK of Ω there exists a level-α

test maximizing (8.2).

[Let β = sup[infΩk Eθϕ(X)], where the supremum is taken over all level-α tests of H : θ ∈ ΩH. Let ϕn be a sequence of level-α tests such that infΩK Eθϕn(X)

tends to β. If ϕni is a subsequence and ϕ a test (guaranteed by Theorem 8.5.1 of the Appendix) such that Eθϕni (X) tends to Eθϕ(X) for all θ ∈ Ω, then ϕ is a level-α test and infΩk Eθϕ(X) = β.]