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Radon-Nikodym Derivatives

  • Michel Simonnet
Part of the Universitext book series (UTX)

Summary

10.1 We define summable families of measures on a same semiring S.

10.2 Let µ be a measure on a semiring S whose underlying set is Ω. A function g: Ω → C is said to be locally µ-integrable whenever gl A a is µ-integrable for every A in S; then the measure gµ: A → ∫ gl A is called the measure with density g relative to µ. V(gµ) = |g|Vµ, (Proposition 10.2.1), and ∫ f dV(gµ) = ∫ f |g| dVµ for all functions f: Ω → [0, +∞] (Theorem 10.2.2). A mapping f from Ω into a Banach space is gµ-measurable (respectively, essentially gµ-integrable) if and only if fg is µ-measurable (respectively, essentially µ-integrable) (Theorem 10.2.1).

When µ is Lebesgue measure on an interval, let us observe that every function continuous on that interval is locally µ-integrable.

10.3 Let µ be a measure on a semiring S. Let ℛ be the ring generated by S. A measure v on S is said to be absolutely continuous with respect to µ if every µ-negligible σ(S)-set is v-negligible. Another equivalent condition is the following condition. For every E in S and for every ɛ > 0, there exists δ > 0 such that for all F in ℛ contained in E and satisfying the inequality |µ|(F) ≤ δ, we have |v|(F) ≤ ɛ (Theorem 10.3.2). This also amounts to saying that v has a density g with respect to µ (Theorem 10.3.1, Radon-Nikodym). Measures µ and v on S are said to be mutually singular whenever inf (|µ|, |v|) = 0. This means that µ and v are concentrated on disjoint sets (Propositions 10.3.3 and 10.3.4). Every measure on S can be written, uniquely, as the sum of a measure gµ absolutely continuous with respect to µ and a measure v such that µ and v are mutually singular (Theorem 10.3.3).

10.4 We combine different operations on the measures.

10.5 We show that L C q (µ) may be regarded as the dual of L C q (µ) (with 1 ≤ p < +∞, q exponent conjugate to p) (Theorems 10.5.1 and 10.5.2). We also characterize those continuous linear functionals on L C (µ) that can be written f ↦ ∫fg dµ for g in L C 1 (µ) (Proposition 10.5.1).

10.6 We describe the dual of L C (µ) (Proposition 10.6.1). A necessary and sufficient condition that this dual be equal to L C 1 (µ) is that Ω be a union of a finite number of atoms and a locally µ-negligible set (Proposition 10.6.2).

Keywords

Riesz Space Metrizable Space Countable Union Linear Isometry Continuous Linear Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Michel Simonnet
    • 1
  1. 1.Department of MathematicsUniversity of DakarSenegal

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