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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Linearity can fail without an integrability hypothesis

Statement refuted

The Lebesgue integral is linear on all measurable real-valued functions.

Facts & Assumptions

Given: The functions f:=χ[0,∞) and g:=−χ[0,∞) on R.

[L1]

Linearity is proved only on L1(μ) (The Lebesgue integral is linear on L1(μ)).

Counterexample

technique · direct
1.1

The sum f+g is the zero function, so its integral is 0.

given
2.1

But f has integral +∞, while g would have to contribute [step 1.1, L1, algebra] ∎ −∞ for any linear identity to hold. Thus ∫(f+g) dλ=0≠+∞+(−∞), so the Statement is false and [L1] cannot be widened beyond L1.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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