DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Weighted total-variation space for population measures

Definition

Fix $a > 0$ , put $E = [a, \infty)$ , and define the weight $w (x) = 1 + x^{2}$ . Let $M_{w} (E)$ be the real vector space of finite signed Borel measures $σ$ on $E$ for which

$∥ σ ∥_{w} : = \int_{E} w (x) ∣ σ ∣ (d x) < \infty,$

where $∣ σ ∣$ is the total-variation measure. The norm is $∥ \cdot ∥_{w}$ . A positive population measure is a nonnegative member $μ \in M_{w} (E)$ . Its number, mass, and second moment are $M_{0} (μ) = \int_{E} 1 d μ$ , $M_{1} (μ) = \int_{E} x d μ$ , and $M_{2} (μ) = \int_{E} x^{2} d μ$ . The space and its completeness are established in The weighted total-variation measure space is complete.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Philippe Laurençot, Weak compactness techniques and coagulation equations, §§1 and 3 (standard reference, not scraped)