DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|)

Let $ν$ be a signed measure or complex measure on $(X, A)$ . Because $∣ ν ∣$ is a measure (The total variation of a signed or complex measure is a positive measure), define $L^{1} (ν) : = L^{1} (∣ ν ∣)$ using the published meaning of $L^{1}$ from The class $L^{1} (μ)$ of integrable functions.

If $f \in L^{1} (ν)$ , choose complex simple functions $s_{n}$ as in Every L^1 function admits dominated complex simple approximations with $\int ∣ f - s_{n} ∣ d ∣ ν ∣ \to 0$ . The simple-integral bound Simple integrals are bounded by total variation makes $(\int s_{n} d ν)$ a Cauchy sequence in $C$ , and its limit is independent of the chosen approximating sequence. Define $\int f d ν : = \lim_{n \to \infty} \int s_{n} d ν .$

For a measurable set $E \in A$ , define $\int_{E} f d ν : = \int f 1_{E} d ν .$

Depends on