Continuous with respect to counting measure

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1. If µ and ν are finite measures on a σ-algebra S, show that ν is absolutely continuous with respect to µ if and only if for every ε > 0 there is a δ > 0 such that for any A ∈ S, µ( A) <>δ implies ν( A) ε.

2. Lebesgue measure λ is absolutely continuous with respect to counting measure c on [0, 1], which is not σ-finite. Show that the conclusion of the Radon-Nikodym theorem does not hold in this case (there is no "derivative" dλ/dc having the properties of h in the Radon-Nikodym theorem).

Reference no: EM13990809

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