TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-17

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Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal

Statement

Let $X$ be a set.

The intersection of every nonempty family of sigma-algebras on $X$ is a sigma-algebra on $X$ .
For every $E \subseteq P (X)$ , the family $Σ_{X} (E)$ of The sigma-algebra generated by a family of sets is nonempty, and $σ_{X} (E)$ is the unique smallest sigma-algebra on $X$ containing $E$ .

Facts & Assumptions

Given: A set $X$ , a nonempty family $S$ of sigma-algebras on $X$ , and a family $E \subseteq P (X)$ , with $Σ_{X} (E)$ and $σ_{X} (E)$ as in The sigma-algebra generated by a family of sets.

Proof

technique · direct

1.1

Every member of $S$ contains $\emptyset$ ; if $A$ belongs to every member, then so does $X ∖ A$ ; and if every $A_{n}$ belongs to every member, then so does $⋃_{n} A_{n}$ . Hence $⋂ S$ is a sigma-algebra on $X$ .

given

1.2

The power set $P (X)$ is a sigma-algebra on $X$ containing $E$ , so $P (X) \in Σ_{X} (E)$ and the defining intersection for $σ_{X} (E)$ is taken over a nonempty family.

givenconstruct

2.1

By step 1.1, $σ_{X} (E)$ is a sigma-algebra. Every set in $E$ belongs to every member of $Σ_{X} (E)$ , so $E \subseteq σ_{X} (E)$ ; and the defining intersection is contained in every sigma-algebra containing $E$ . Thus it is the unique smallest such sigma-algebra.

step 1.1step 1.2∎

Depends on

The sigma-algebra generated by a family of sets

Used by

Cited to discharge well-definedness by The sigma-algebra generated by a family of sets.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 3 results over 3 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, Lemma 2.7 (standard reference, not scraped)
T. Tao, An Introduction to Measure Theory, Exercises 1.4.13-1.4.14 (standard reference, not scraped)