DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedaudited 2026-09-10

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.

Levy prokhorov metric

Definition

For Borel probabilities $μ$ , $ν$ on a metric space S, put $F^{[ε]} = {x : d (x, F) \leq ε}$ for nonempty closed F, and $\emptyset^{[ε]} = \emptyset$ . Define $π (μ, ν)$ as the infimum of $ε$ >0 such that, for every closed F, both $μ (F) \leq ν (F^{[ε]}) + ε$ and $ν (F) \leq μ (F^{[ε]}) + ε$ . The admissible set contains every $ε$ >=1 and is bounded below by zero, so its real infimum exists by Every nonempty set bounded below has an infimum. Enlargements are closed because distance to a nonempty set is continuous. $∣ d (x, A) - d (y, A) ∣ \leq d (x, y)$ , so the distance to a fixed nonempty set is $1$ -Lipschitz The metric assertion is proved in the following lemma.

Depends on

Used by

Levy prokhorov distance is a metric Lemma

Dependency tree · two levels

22 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

van Gaans, §4, pp. 9–10 (open-enlargement convention; equivalence must be proved locally) (standard reference, not scraped)