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

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

Statement

For every algebra $A$ of subsets of $X$ ,

$m_{X} (A) = σ_{X} (A) .$

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]

The family $M$ is closed under complements (The monotone class generated by an algebra is closed under complements) and binary intersections (The monotone class generated by an algebra is closed under finite intersections).

[L3]

An algebra closed under increasing countable unions is a sigma-algebra (An algebra closed under increasing countable unions is a sigma-algebra).

[L4]

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

Proof

technique · direct

1.1

By [L2], $M$ contains $X$ and is closed under complements and binary intersections, hence under finite unions; it is therefore an algebra.

L1L2algebra

1.2

Every sigma-algebra is closed under increasing unions and, by De Morgan's law, decreasing intersections. Thus $σ_{X} (A)$ is a monotone class containing $A$ , and minimality in [L1] gives $M \subseteq σ_{X} (A)$ .

L1L4algebra

2.1

Since $M$ is a monotone class by [L1], it is closed under increasing countable unions. Step 1.1 and [L3] make it a sigma-algebra containing $A$ , so [L4] gives $σ_{X} (A) \subseteq M$ .

step 1.1L1L3L4

3.1

The inclusions of steps 2.1 and 1.2 prove the equality.

step 2.1step 1.2∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 5 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, Theorem 2.10 (standard reference, not scraped)
A. Dembo, Probability Theory lecture notes, Theorem 1.1.44 (standard reference, not scraped)