TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

Generated sigma-algebras are monotone in their generators and idempotent

Statement

For families $E, F \subseteq P (X)$ :

if $E \subseteq F$ , then $σ_{X} (E) \subseteq σ_{X} (F)$ ;
$σ_{X} (σ_{X} (E)) = σ_{X} (E)$ .

Facts & Assumptions

Given: Families $E, F \subseteq P (X)$ .

[L1]

The family $σ_{X} (E)$ is the unique smallest sigma-algebra on $X$ containing $E$ (Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal).

Proof

technique · direct

1.1L1

If $E \subseteq F$ , then the sigma-algebra $σ_{X} (F)$ contains $E$ , so minimality in [L1] gives $σ_{X} (E) \subseteq σ_{X} (F)$ .

2.1L1∎

The family $σ_{X} (E)$ is already a sigma-algebra. It is therefore the smallest sigma-algebra containing itself, and [L1] gives $σ_{X} (σ_{X} (E)) = σ_{X} (E)$ .

Depends on

Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal

Used by

Two families generate the same sigma-algebra when each lies in the sigma-algebra generated by the other Theorem

Dependency tree · two levels

2 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

R. F. Bass, Real Analysis for Graduate Students, version 5.0, Section 2.1 (standard reference, not scraped)