TheoremStatement: AI-adaptedProof: 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.

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

Statement

Let $E, F \subseteq P (X)$ . If

$E \subseteq σ_{X} (F) and F \subseteq σ_{X} (E),$

then $σ_{X} (E) = σ_{X} (F)$ .

Facts & Assumptions

Given: Families $E, F \subseteq P (X)$ satisfying the two displayed inclusions.

[L1]

Generated sigma-algebras are monotone in their generators and idempotent (Generated sigma-algebras are monotone in their generators and idempotent).

Proof

technique · direct

1.1givenL1

From $E \subseteq σ_{X} (F)$ , monotonicity and idempotence in [L1] give $σ_{X} (E) \subseteq σ_{X} (σ_{X} (F)) = σ_{X} (F)$ .

2.1step 1.1givenL1∎

Interchanging $E$ and $F$ gives $σ_{X} (F) \subseteq σ_{X} (E)$ ; together with step 1.1 this proves equality.

Depends on

Generated sigma-algebras are monotone in their generators and idempotent

Used by

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

T. Tao, An Introduction to Measure Theory, Exercise 1.4.14 (standard reference, not scraped)