LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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

1.1

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 $C_{B}$ .

L1L2algebra

2.1

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

step 1.1L1∎

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 5 results over 4 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)