LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Approximation in symmetric difference by a generating algebra

Statement

If $μ (X) < \infty$ and an algebra $C$ of subsets of X generates $A$ , then for every $E \in A$ and $ε > 0$ there is $C \in C$ with $μ (E △ C) < ε$ .

Facts & Assumptions

[F1]

An algebra contains X and is closed under complements and finite unions Algebras of subsets.

[F2]

The measure of an increasing union is the supremum of the measures Continuity from below for measures.

[F3]

Union errors are bounded by the sum of their measures Finite and countable subadditivity of measures.

[F4]

The generated sigma-algebra is the smallest sigma-algebra containing its generators Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal.

Proof

Given: The objects and hypotheses in the statement.

1.1F1

Let $D = {E \in A : for every ε > 0 some C \in C has μ (E △ C) < ε}$ . Every $C \in C$ lies in $D$ by using itself. For complements, $(X ∖ E) △ (X ∖ C) = E △ C$ , and $X ∖ C \in C$ . Thus $D$ contains X and is closed under complements.

2.1F1F2F3step 1.1

Let $E_{j} \in D$ and $E = ⋃_{j \geq 1} E_{j}$ . For $H_{m} = ⋃_{j = 1}^{m} E_{j}$ , continuity from below gives $μ (H_{m}) ↑ μ (E) < \infty$ . By additivity on $E = H_{m} ⊔ (E ∖ H_{m})$ , some $m \geq 1$ satisfies $μ (E ∖ H_{m}) < ε / 2$ . Choose finitely many $C_{j} \in C$ with $μ (E_{j} △ C_{j}) < ε / (2 m)$ . Then $C = ⋃_{j = 1}^{m} C_{j} \in C$ and $μ (E △ C) \leq μ (E ∖ H_{m}) + \sum_{j = 1}^{m} μ (E_{j} △ C_{j}) < ε$ .

3.1F4step 1.1step 2.1∎

Thus $D$ is a sigma-algebra containing $C$ . By the minimality of $σ (C) = A$ , $A \subseteq D$ , and the reverse inclusion is built into its definition. This proves the approximation assertion.

Depends on

Dependency tree · 2 levels

Sources

E–W Proposition 2.15 proof and Exercise 2.7.3