ExampleConstruction: 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.

The sigma-algebra generated by a two-step simple function

Example

Let $\emptyset \neq A \neq X$ , let $α \neq β$ , and define

$f : = α 1_{A} + β 1_{X ∖ A} .$

Then $f$ takes exactly the two values $α$ and $β$ , and

$σ (f) = {\emptyset, X, A, X ∖ A} .$

Facts & Assumptions

Given: A set $X$ , a subset $A$ with $\emptyset \neq A \neq X$ , distinct real numbers $α \neq β$ , and the two-step function $f = α 1_{A} + β 1_{X ∖ A}$ .

[L1]

The sigma-algebra generated by a function is the collection of its Borel preimages. (The sigma-algebra generated by a function)

Verification

technique · direct

1.1

Because $f$ takes only the two values $α$ and $β$ , every preimage [L1, given] $f^{- 1} (B)$ is one of $\emptyset$ , $X$ , $A$ , or $X ∖ A$ . So $σ (f)$ is contained in that four-set family.

L1given

2.1

Since $α \neq β$ , choose a real threshold strictly between them. The [step 1.1, L1] corresponding open ray has preimage $A$ or $X ∖ A$ , and the other three sets arise as in the previous example from $R$ and $\emptyset$ . Hence all four sets belong to $σ (f)$ , so they are exactly its members.

step 1.1L1∎

Depends on

The sigma-algebra generated by a function

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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, Section 3.1 (standard reference, not scraped)