Measures and Probabilities pp 193-223 | Cite as

# Radon-Nikodym Derivatives

## 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 *g*l_{ A } a is µ-integrable for every *A* in *S*; then the measure *g*µ: *A* → ∫ *g*l_{ A } is called the measure with density *g* relative to µ. *V*(*g*µ) = |*g*|*V*µ, (Proposition 10.2.1), and ∫^{•} *f dV*(*g*µ) = ∫^{•} *f* |*g*| *dV*µ for all functions *f:* Ω → [0, +∞] (Theorem 10.2.2). A mapping *f* from Ω into a Banach space is *g*µ-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 *g*µ 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* *d*µ 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

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