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

Suppose that the problem of testing θ ∈ H against θ ∈ K 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 infK [w(θ )Eθϕ(X) + u(θ )] for any weight functions w(θ ) ≥ 0, u(θ )that are invariant under G¯ .