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.
The completion domain and proposed completed set function of a measure space
Definition
Let be a measure space. Its completion domain is
For a displayed representation , define the proposed completed set function by
The value must not depend on the representation. That obligation is discharged by The completed measure is independent of the representing measurable set ↗, and Assuming countable choice, the completion domain is a sigma-algebra ↗ proves, under the Axiom of Countable Choice (The Axiom of Countable Choice ()), that is a sigma-algebra. Thus the definite phrases completion sigma-algebra and completed measure are used only with those two results in force. Under the same choice hypothesis, Assuming countable choice, every measure space has a unique complete extension to its completion ↗ proves that is the unique complete extension on this domain.
Depends on
Used by
- L(ℝⁿ) is exactly the completion of the restriction of λₙ to the Borel sets Corollary
- Boldface Sigma-one-three measurability Definition
- The completed product measure Definition
- A square-integrable separable product kernel Example
- Assuming choice, the completion of the Borel Dirac measure at zero is defined on every subset of the real line Example
- Assuming countable choice, the completion domain is a sigma-algebra Lemma
- The completed measure is independent of the representing measurable set Lemma
- Compositions, iterates and completions preserve invariance Proposition
- A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra Theorem
- Assuming countable choice, every measure space has a unique complete extension to its completion Theorem
- Assuming countable choice, the Carathéodory domain is the completion of the sigma-finite extension Theorem
- Every integer-base circle map is strongly mixing Theorem
- Measurable integration extends smooth density integration Theorem
- Positive smooth densities give Radon volume Theorem
- Weak mixing is equivalent to absence of nonconstant eigenfunctions Theorem
Dependency tree · two levels
9 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., Theorem 1.9 (standard reference, not scraped)