TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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Sigma-algebras are closed under countable intersections, differences, symmetric differences, and set limits

Statement

Let $A$ be a sigma-algebra on $X$ . Then $A$ is closed under countable intersections, differences, and symmetric differences. If every term of a sequence $(A_{n})_{n \in N}$ lies in $A$ , then both ${lim inf}_{n} A_{n}$ and ${lim sup}_{n} A_{n}$ (Limit superior and limit inferior of a sequence of sets) lie in $A$ .

Facts & Assumptions

Given: A sigma-algebra $A$ on $X$ and a sequence $(A_{n})_{n \in N}$ in $A$ .

[L1]

A sigma-algebra contains the empty set and is closed under complements and countable unions (Sigma-algebras).

[L2]

Set liminf is a countable union of tail intersections, and set limsup is a countable intersection of tail unions (Limit superior and limit inferior of a sequence of sets).

Proof

technique · direct

1.1L1algebra

De Morgan's identity gives $⋂_{n} A_{n} = X ∖ ⋃_{n} (X ∖ A_{n}) \in A$ by [L1]. Finite and empty intersections are included by repeating terms and by $⋂ \emptyset = X = X ∖ \emptyset$ .

2.1step 1.1L1algebra

If $A, B \in A$ , then $A ∖ B = A \cap (X ∖ B) \in A$ by step 1.1, and $A △ B = (A ∖ B) \cup (B ∖ A) \in A$ .

3.1step 1.1L1L2∎

Each tail intersection and tail union of $(A_{n})$ belongs to $A$ by step 1.1 and [L1]. Applying countable union and intersection closure once more to the formulas in [L2] puts both ${lim inf}_{n} A_{n}$ and ${lim sup}_{n} A_{n}$ in $A$ .

Depends on

Used by

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, Definition 2.1 and Exercise 2.9 (standard reference, not scraped)