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|)

Definition

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

Used by

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard F. Bass, Real Analysis for Graduate Students, Chapter 12 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, Chapter 9A (standard reference, not scraped)