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 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 ^q_C (µ) may be regarded as the dual of L ^q_C (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 ¹_C (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 ¹_C (µ) is that Ω be a union of a finite number of atoms and a locally µ-negligible set (Proposition 10.6.2).