Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: 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.

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A pointwise limit of integrable functions need not be integrable

Statement refuted

Every pointwise limit of integrable functions is integrable.

Facts & Assumptions

Given: Counting measure on N and the functions fn=χ{0,…,n}.

[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).

Counterexample

technique · direct
1.1

Each fn has finite support, so it is integrable on [L1, L2, given] (N,P(N),#).

2.1

For every k∈N, one has fn(k)→1. The pointwise limit is the [step 1.1, L1, L2, algebra] ∎ constant function 1, whose integral under counting measure is +∞, so it is not integrable by [L2]. Thus the Statement is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources