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Let An be an increasing sequence of σ-algebras whose union generates a σ-algebra S. Let µ and ν be probability measures on S such that ν is singular with respect to µ on S, but absolutely continuous with respect to µ on each An with Radon-Nikodym derivative fn . Show that fn → 0 a.s. for µ. Hint: If fn → f a.s., show by Fatou's lemma that (A f dµ ≤ ν( A) for every A ∈ S.

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