LemmaStatement: 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.

The monotone class generated by an algebra is closed under complements

Statement

If $A$ is an algebra of subsets of $X$ , then the generated monotone class $m_{X} (A)$ is closed under complements relative to $X$ .

Facts & Assumptions

Given: An algebra $A$ on $X$ and $M : = m_{X} (A)$ .

[L1]

An algebra is closed under complements (Algebras of subsets).

[L2]

The family $M$ is the smallest monotone class containing $A$ (The generated monotone class exists and is minimal).

Proof

technique · direct

1.1L2algebra

Put $C : = {E \in M : X ∖ E \in M}$ . If $(E_{n})$ increases in $C$ , then $X ∖ E_{n}$ decreases, so monotone closure of $M$ places both $⋃_{n} E_{n}$ and its complement $⋂_{n} (X ∖ E_{n})$ in $M$ . The decreasing case is the same with union and intersection interchanged. Hence $C$ is a monotone class.

2.1step 1.1L1L2∎

If $E \in A$ , then $X ∖ E \in A \subseteq M$ by [L1] and [L2], so $A \subseteq C$ . Minimality in [L2] gives $M \subseteq C$ , which is the asserted complement closure.

Depends on

Used by

The monotone class generated by an algebra equals the sigma-algebra it generates Theorem

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

R. F. Bass, Real Analysis for Graduate Students, version 5.0, proof of Theorem 2.10 (standard reference, not scraped)