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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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

1.1

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 $⋂ \emptyset = X = X ∖ \emptyset$ .

L1algebra

2.1

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

step 1.1L1algebra

3.1

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 $A$ .

step 1.1L1L2∎

Depends on

Used by

F-sigma and G-delta subsets of the real line are Borel Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 3 results over 2 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, Definition 2.1 and Exercise 2.9 (standard reference, not scraped)