ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Indicator of the rationals has zero essential supremum but pointwise supremum one

Example

On $[0, 1]$ with Lebesgue measure, let $f : = χ_{Q \cap [0, 1]}$ . Then

$\sup_{x \in [0, 1]} f (x) = 1, ∥ f ∥_{\infty} = 0.$

So essential supremum and pointwise supremum need not agree.

Facts & Assumptions

Given: The function $f = χ_{Q \cap [0, 1]}$ on $[0, 1]$ .

[L1]

The essential supremum is the infimum of the essential bounds (The essential supremum of a measurable function with respect to a measure).

[L2]

$Q$ is countable, and every at most countable subset of $R$ is null ( $Q$ is countably infinite, Every at most countable subset of $R$ has measure zero).

Verification

technique · Use that the rationals in $[0,1]$ are countable and therefore null, so every positive threshold is exceeded only on a null set

1.1

The pointwise supremum is $1$ because $f (x) = 1$ on every rational point of $[0, 1]$ .

given

1.2

If $0 < ε < 1$ , then ${f > ε} = Q \cap [0, 1]$ , which is null by [L2]; if $ε \geq 1$ , then ${f > ε} = \emptyset$ , which is also null. Hence every $ε > 0$ is an essential bound in the sense of [L1], and therefore $∥ f ∥_{\infty} = 0$ .

L1L2

2.1

Steps 1.1 and 1.2 prove the two claims.

step 1.1step 1.2∎

Depends on

Used by

FALSE: the essential supremum equals the pointwise supremum False statement

Dependency tree · two levels

34 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

John K. Hunter, Measure Theory, Definition 7.3 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., Section 2.5 (standard reference, not scraped)