Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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 an indicator function

Example

If f=1A on X, then

σ(f)={∅, X, A, X∖A}.

Facts & Assumptions

Given: A set X, a subset A⊆X, and the indicator function f=1A.

[L1]

The sigma-algebra generated by a function is σ(f)={f−1(B):B∈B(R)}. (The sigma-algebra generated by a function)

Verification

technique · direct
1.1L1given

Because f takes only the values 0 and 1, every preimage f−1(B) is [L1, given] one of ∅, X, A, or X∖A. So σ(f) is contained in that four-set family.

2.1step 1.1L1∎

The four sets all occur as preimages: [step 1.1, L1] A=f−1((1/2,∞)), X∖A=f−1((−∞,1/2)), X=f−1(R), and ∅=f−1(∅). Therefore they are exactly the members of σ(f).

Depends on

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