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 finite intersections

Statement

If $A$ is an algebra on $X$ , then $m_{X} (A)$ is closed under binary intersections.

Facts & Assumptions

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

[L1]

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

[L2]

Every member of $M$ intersects every member of $A$ in a member of $M$ (Every member of the generated monotone class intersects every original algebra member inside the generated class).

Proof

technique · direct

1.1L1L2algebra

Fix $B \in M$ and put $C_{B} : = {E \in M : E \cap B \in M}$ . As before, intersection with $B$ commutes with increasing unions and decreasing intersections, so $C_{B}$ is a monotone class. By symmetry of intersection and [L2], every $E \in A$ lies in $C_{B}$ .

2.1step 1.1L1∎

Minimality in [L1] gives $M \subseteq C_{B}$ . Since $B$ was arbitrary, $E \cap B \in M$ for all $E, B \in M$ .

Depends on

Used by

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

Dependency tree · two levels

4 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)