1 let micro and nu be finite measures such that nu is, Ask an Expert

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1. Let µ and ν be finite measures such that ν is absolutely continuous with respect to µ, and f = dν/dµ. Show that for each r > 0, the Hahn decomposition of ν - rµ gives sets A and Ac such that f ≥ r a.e. on A and f ≤ r a.e. on Ac .

2. Prove the Radon-Nikodym theorem, for finite measures, from the Hahn decomposition theorem. Hint: See Problem 8 on how to define f.

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