Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated
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 bounds bounded-Lipschitz discrepancy

Statement

For μ,ν∈Mw(E),

dBL(μ,ν)≤∥μ−ν∥TV≤∥μ−ν∥w.

Facts & Assumptions

Given: The weight w(x)=1+x2≥1 and the test class in the bounded-Lipschitz discrepancy.

[F1]

Bounded-Lipschitz discrepancy for finite measures: every admissible test has ∥φ∥∞≤1.

[F2]

Weighted total-variation space for population measures: the weighted norm is ∫w d∣σ∣ with w≥1.

Proof

1.1givenF1

For each admissible φ, ∣∫φ d(μ−ν)∣≤∫∣φ∣ d∣μ−ν∣≤∥μ−ν∥TV by [F1]. Taking the supremum proves the first inequality.

2.1F2algebra∎

Since w≥1, integrating w against ∣μ−ν∣ dominates its total mass, which is the total-variation norm.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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Sources