Suppose that P and Q are two probability measures on a measurable space (Ω,F).
Definition A.10. Q is said to be absolutely continuous with respect to P on the σ-algebra F, and we write Q P, if for all A ∈ F,
If both Q P and P Q hold, we will say that Q and P are equivalent, and we will write Q ≈ P.
The following characterization of absolute continuity is known as the Radon–Nikodym theorem:
Theorem A.11 (Radon–Nikodym). Q is absolutely continuous with respect to P on F if and only if there exists an F-measurable function φ ≥ 0 such that
for all F-measurable functions F ≥ 0.
Proof. See, e.g., §17 ...