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 translated unit intervals show that Egorov needs finite total measure
Statement refuted
Egorov's theorem holds on every measure space.
Facts & Assumptions
Given: Lebesgue measure on and the sequence .
This sequence converges almost everywhere to but not in measure. (The translates of the unit interval converge almost everywhere to zero but not in measure)
Almost-uniform convergence implies convergence in measure. (Almost uniform convergence implies almost-everywhere convergence and convergence in measure)
Counterexample
By [L1], the sequence converges almost everywhere to .
If this convergence were almost uniform, then [L2] would force convergence in measure as well. But [L1] says that convergence in measure fails. So the convergence is not almost uniform.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
6 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.