Suppose that the problem of testing H against K remains invariant under G,

Suppose that the problem of testing θ ∈ ΩH against θ ∈ ΩK remains invariant under G, that there exists a UMP almost invariant test ϕ0 with respect to G, and that the assumptions of Theorem 8.5.1 hold. Then ϕ0 maximizes infΩK [w(θ)Eθϕ(X) + u(θ)] for any weight functions w(θ) ≥ 0, u(θ)

that are invariant under G¯.