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 in L^1(mu)
Definition
Let be a measure space and let (The class of integrable functions). The sequence converges to in when
On this page is still the class of integrable representatives rather than the quotient by almost-everywhere equality, so the expression above is read on the functions themselves.
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 typewriter sequence converges in measure and in L¹ but nowhere pointwise Example
- FALSE: convergence in L¹(mu) forces almost-everywhere convergence False statement
- Convergence in L¹(mu) implies convergence in measure Theorem
- Vitali convergence theorem on finite and sigma-finite measure spaces 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., Proposition 2.29 (standard reference, not scraped)
- Terence Tao, 245A Notes 4: Modes of convergence (standard reference, not scraped)