DefinitionDefinition: Literature-sourcedProof: 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.
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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 for probability measures

Definition

Let $d \geq 1$ , let $W : R^{d} \to [1, \infty)$ be Borel measurable, and let $μ, ν$ be Borel probability measures on $R^{d}$ with $\int W d μ < \infty$ and $\int W d ν < \infty$ . Define their $W$ -weighted total-variation distance by $∥ μ - ν ∥_{W} : = \sup_{f B o r e l, ∣ f ∣ \leq W} ∣ \int f d μ - \int f d ν ∣ .$ The moment hypotheses make every integral in the supremum finite. This is the signed-measure test-function norm used in weighted-variation bounds. For $W \equiv 1$ it has the usual factor of two relative to the event-supremum probability distance in Total variation distance for probability laws. Here a Borel probability measure is in the sense of Probability measures and probability spaces, on the Borel sigma-algebra of The Borel sigma-algebra of a topological space.

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Used by

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Sources

Alain Durmus and Éric Moulines, Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm, §1, notation and conventions (standard reference, not scraped)
Gautam Iyer, Lecture 20: Langevin Monte Carlo, course notes, §3 (standard reference, not scraped)