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.
Measurability is relative to sigma-algebras on both domain and codomain
In a measurable space Measurable spaces and measurable sets, the word measurable never belongs to a function by itself. It belongs to a function together with a source sigma-algebra and a target sigma-algebra. On this page the target on or is always the Borel sigma-algebra, not the Lebesgue sigma-algebra.
That convention is load-bearing. A Lebesgue measurable function on means
not a map into . The distinction is
exactly what makes the later counterexample
g \circ f with merely Lebesgue measurable possible.
Depends on
Used by
Nothing in the library uses this result yet.
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
- Sheldon Axler, Measure, Integration and Real Analysis, Section 2B (standard reference, not scraped)
- John K. Hunter, Measure Theory, Chapter 3 (standard reference, not scraped)