Based on X with distribution indexed by , the problem is to test

Based on X with distribution indexed by θ ∈ , the problem is to test θ ∈ ω versus θ ∈ ω

. Suppose there exists a test φ such that Eθ[φ(X)] ≤ β for all θ in ω, where β < α. Show there exists a level α test φ∗(X) such that Eθ[φ(X)] ≤ Eθ[φ∗(X)] , for all θ in ω and this inequality is strict if Eθ[φ(X)] < 1.