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. ...

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