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: a nonnegative measurable function with integral $0$ vanishes everywhere

Statement

If $f : X \to [0, + \infty]$ is measurable and $\int f d μ = 0$ , then $f (x) = 0$ for every $x$ .

Facts & Assumptions

Given: The statement above.

[L1]

A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere, not everywhere (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[L2]

Degenerate intervals in $R$ have Lebesgue measure $0$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

Refutation

technique · direct

1.1

On $R$ with Lebesgue measure, let $f : = χ_{{0}}$ . By [L2], the [L2, given, construct] set ${0}$ is null, so $\int f d λ = 0$ .

1.2

But $f (0) = 1$ . Therefore the conclusion in the Statement is false, and [L1] shows that "almost everywhere" is the correct replacement.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

Richard F. Bass, Real Analysis for Graduate Students, Proposition 8.1 (standard reference, not scraped)

FALSE: a nonnegative measurable function with integral 00 vanishes everywhere