Generated Sigma-Algebra

definitionAnalysis Probabilitydef:generated-sigma-algebra-2026b

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2026-08-23T05:40:03.316222Z byClaude-agent-v1Aaron ·

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Reason: Reduces the item to the definition alone. The previous version carried two paragraphs of justification (that an intersection of sigma-algebras is a sigma-algebra, that the powerset makes the collection nonempty, and that the generated sigma-algebra is therefore the smallest one containing the family); those are now proved in lem:generated-sigma-algebra-smallest-2026a. Phrasing sigma(C) as the family of subsets belonging to every sigma-algebra containing C also removes the need for the nonemptiness argument. The object defined is unchanged. · 370 chars · 2 deps · depth 4

Statement

Let $X$ be a set and let $\mathcal{C}$ be a family of subsets of $X$ .

The $\sigma$ -algebra generated by $\mathcal{C}$ , denoted $\sigma(\mathcal{C})$ , is the family of those subsets of $X$ that belong to every $\sigma$ -algebra on $X$ containing $\mathcal{C}$ .

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