Suppose that P and Q are two probability measures on (Ω,F), and denote by
the Lebesgue decomposition of P with respect to Q as in Theorem A.17. For fixed c ≥ 0, we let
where we make use of the convention that dP/dQ = ∞on N.
Proposition A.33 (Neyman–Pearson lemma). If A ∈ F is such that Q[ A ]≤ Q[ A0 ], then P[ A ]≤ P[ A0 ].
Proof. Let Then F ≥ 0 on N, and (dP/dQ − c)F ≥ 0. Hence
This proves the proposition.
Remark A.34. ...