Alphabeta MathHow 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.
Integral probability metrics
Definition
Let be a measurable space, let and be probability measures on it, and let be a nonempty class of real-valued -measurable functions that are integrable under both and . The integral probability metric induced by is It is nonnegative and symmetric, and it may equal ; no separation property is asserted for an arbitrary class .
Depends on
- Probability measures and probability spaces
- Integrable real and complex functions, and their integrals
- The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined
- Every subset of $\overline{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\overline{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$
Used by
Dependency tree · two levels
19 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
- Martin Arjovsky, Soumith Chintala and Léon Bottou, Wasserstein GAN, §3, equation (2), PDF pp. 5–6 (standard reference, not scraped)
- Filippo Santambrogio, Introduction to Optimal Transport Theory, §§1.1–1.3, PDF pp. 3–7 (standard reference, not scraped)