CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)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.

On a finite measure space, convergence in measure has an almost-uniformly convergent subsequence

Statement

Let $(X, A, μ)$ be a measure space with $μ (X) < + \infty$ , and let $f_{n}, f : X \to R$ be measurable. If $f_{n} \to f$ in measure, then some subsequence of $(f_{n})$ converges to $f$ almost uniformly.

Facts & Assumptions

Given: A finite measure space $(X, A, μ)$ and measurable functions $f_{n}, f : X \to R$ such that $f_{n} \to f$ in measure.

[L1]

Convergence in measure has a subsequence converging almost everywhere to the same limit. (Riesz's subsequence theorem for convergence in measure)

[L2]

On a finite measure space, almost-everywhere convergence implies almost-uniform convergence. (Egorov's theorem)

Proof

technique · direct

1.1

By [L1], there is a subsequence $(f_{n_{k}})$ converging to $f$ almost everywhere.

2.1

Apply [L2] to the subsequence from step 1.1. It converges to $f$ almost uniformly.

step 1.1L2∎

Depends on

Used by

Dependency tree · two levels

8 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., Theorems 2.30 and 2.33 (standard reference, not scraped)