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2Midterm2 - Analysis Midterm 2 Write solutions in a neat...

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Analysis Midterm 2 Write solutions in a neat and logical fashion, giving complete reasons for all steps. 1. Let (Ω , F ) be a measurable space on which λ, μ and ν are (finite) measures. Show that if ν μ and μ λ then ν λ ; further, carefully verify the relation = among Radon-Nikodym derivatives (which should be made precise). Solution To see that ν λ let C ∈ F with λ ( C ) = 0: as μ λ it follows that μ ( C ) = 0; as ν μ it follows that ν ( C ) = 0. Let (0 ) f (= ) ∈ L , F , λ ) and (0 ) g (= ) ∈ L , F , μ ) be Radon- Nikodym derivatives as indicated, so that ν = μ g and μ = λ f . The indicated relation may be expressed by saying that the product gf is a Radon-Nikodym derivative of ν by λ : that is, ν = λ gf ; explicitly, if C ∈ F then ν ( C ) = C gfdλ. To justify this, ν ( C ) = C gdμ = C gdλ f = C gfdλ where Theorem 7.2 is used at the last step. Note that an R-N derivative is defined uniquely up to null sets.
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