Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27
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.

FALSE: pointwise limits of integrable functions are integrable

Statement

Whenever integrable functions fn converge pointwise to f, the limit f is integrable.

Facts & Assumptions

Given: The statement above.

[L1]

Counting measure is a measure on N (Counting measure on an arbitrary set, Counting measure is a measure).

[L2]

Integrability means finiteness of the integral of the modulus (Integrable real and complex functions, and their integrals).

Refutation

technique · direct
1.1

On (N,P(N),#), let [L1, L2, given, construct] fn:=χ{0,…,n}. Each fn is integrable because it has finite support, and fn(k)→1 for every k∈N.

2.1

The pointwise limit is the constant function 1, whose counting-measure [step 1.1, L1, L2, algebra] ∎ integral is +∞, so it is not integrable by [L2]. Therefore the Statement is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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