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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Convergence almost everywhere relative to a measure
Definition
Let be a measure space and let be measurable. The sequence converges to -almost everywhere when there is a measurable -null set such that for every the real sequence converges to .
Equivalently, converges to -almost everywhere when the pointwise convergence statement holds -almost everywhere in the sense of Measure-null sets and almost-everywhere statements relative to a measure.
Depends on
Used by
- The spikes k chi_(0,1/k) converge almost everywhere and in measure to zero but not in L¹ Counterexample
- The translates of the unit interval converge almost everywhere to zero but not in measure Counterexample
- The typewriter sequence converges in measure and in L¹ but nowhere pointwise Example
- FALSE: almost-everywhere convergence implies convergence in measure on every measure space False statement
- FALSE: convergence in L¹(mu) forces almost-everywhere convergence False statement
- FALSE: convergence in measure implies almost-everywhere convergence False statement
- Almost uniform convergence implies almost-everywhere convergence and convergence in measure Theorem
- Egorov's theorem Theorem
- On a finite measure space, almost-everywhere convergence implies convergence in measure Theorem
- Riesz's subsequence theorem for convergence in measure 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
- Gerald B. Folland, Real Analysis, 2nd ed., Section 2.4 (standard reference, not scraped)
- Terence Tao, 245A Notes 4: Modes of convergence (standard reference, not scraped)