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.
Complete measure spaces
Definition
A measure space is complete if every subset of every measurable -null set is measurable: whenever , , and , one has (Measure-null sets and almost-everywhere statements relative to a measure).
Depends on
Used by
- A nonmeasurable subset of a null line shows that the product of complete measures need not be complete Counterexample
- PMEA and PMEA-sigma Definition
- The completion domain and proposed completed set function of a measure space Definition
- FALSE: every subset of a measure-null set is measurable False statement
- Null sets are closed under countable unions and, in a complete space, under arbitrary subsets Proposition
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume Theorem
- Carathéodory's theorem: measurable sets form a sigma-algebra carrying a complete measure Theorem
- On a complete measure space, equality almost everywhere preserves measurability Theorem
- Pettis measurability criterion for strong measurability Theorem
Dependency tree · two levels
2 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
- G. Folland, Real Analysis, 2nd ed., §1.3 (standard reference, not scraped)