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Stochastic Finance, 4th Edition by Alexander Schied, Hans Föllmer

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A.2 Absolutely continuous probability measures

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 RadonNikodym theorem:

Theorem A.11 (RadonNikodym). 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 ...

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