RemarkRemark: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-28

How 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.

Implication table for the main modes of convergence on a finite measure space

On a finite measure space, the page establishes the following implication picture for the four main convergence modes it develops.

from	to	status	reason or witness
almost uniform	almost everywhere	proved	Almost uniform convergence implies almost-everywhere convergence and convergence in measure
almost uniform	in measure	proved	Almost uniform convergence implies almost-everywhere convergence and convergence in measure
almost everywhere	in measure	proved	On a finite measure space, almost-everywhere convergence implies convergence in measure
in $L^{1}$	in measure	proved	Convergence in L^1(mu) implies convergence in measure
in measure + uniform integrability	in $L^{1}$	proved	Vitali convergence theorem on finite and sigma-finite measure spaces
in measure	almost everywhere	false	the typewriter sequence on $[0, 1]$
in $L^{1}$	almost everywhere	false	the same typewriter sequence
in measure	in $L^{1}$	false	the spikes $f_{0} = 0$ and $f_{n} = n χ_{(0, 1 / n)}$ for $n \geq 1$
almost everywhere	in $L^{1}$	false	the same spike sequence

Two companion observations matter just as much as the table.

Convergence in measure on a finite measure space is exactly convergence in the truncated metric of On a finite measure space, the truncated L^1 metric metrises convergence in measure.
Even when convergence in measure does not imply almost-everywhere convergence of the whole sequence, Riesz gives an almost-everywhere convergent subsequence and, on finite measure spaces, Egorov upgrades that subsequence to almost uniform convergence (On a finite measure space, convergence in measure has an almost-uniformly convergent subsequence).

The companion examples page still spells out the same concrete witnesses: the typewriter sequence, the translated unit intervals on $R$ , the spike family $f_{0} = 0$ , $f_{n} = n χ_{(0, 1 / n)}$ for $n \geq 1$ , the explicit Egorov core for $x^{k}$ , the Dirichlet-function Lusin core, and the uniformly integrable but non-dominated disjoint-spike family.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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)
Richard F. Bass, Real Analysis for Graduate Students, Sections 5.3 and 7 (standard reference, not scraped)