Conclusion of the radon-nikodym theorem

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1. Take the Cantor function g defined in the proof of Proposition 4.2.1. As a nondecreasing function, it defines a measure ν on [0, 1] with ν((a, b]) = g(b) - g(a) for 0 ≤ a b ≤ 1 by Theorem 3.2.6. Show that ν ⊥ λ where λ is Lebesgue measure.

2. In [0, 1] with Borel σ-algebra B, let µ( A) = 0 for all A ∈ B of first cate- gory and µ( A) = +∞ for all other A ∈ B. Show that µ is a measure, and that if ν is a finite measure on B absolutely continuous with respect to µ, then ν ≡ 0. Hint: See §3.4, Problems 7-8. Thus, the conclusion of the Radon-Nikodym theorem holds with h ≡ 0, although µ is not σ-finite.

Reference no: EM13990810

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