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A σ-algebra on a set X is a collection of subsets closed under countable unions and complements, containing ∅ and X itself. This closure structure ensures that countable operations preserve measurability. σ-algebras generalize the Borel σ-algebra on ℝ to arbitrary spaces.
Start by verifying that ℝ's Borel σ-algebra satisfies closure properties. Then construct σ-algebras generated by specific collection of sets.
A σ-algebra is not closed under all unions—only countable ones. Uncountable unions of measurable sets need not remain measurable.
From your study of topological spaces, you know that a topology on X is a collection of subsets (the open sets) closed under arbitrary unions and finite intersections. A σ-algebra imposes a different closure discipline: closed under countable unions and complements, with the whole set X and the empty set included. The key word is *countable* — this is precisely the structure needed to assign consistent numerical sizes (measures) to sets without running into contradictions.
Why countable and not arbitrary? The reason is probabilistic and measure-theoretic coherence. If you want to say "the probability of event A or B or C or ..." for a countable list of mutually exclusive events, you need the union to be measurable. If you allowed arbitrary (uncountable) unions, you could assemble unmeasurable sets from measurable atoms — a breakdown that leads to paradoxes like the Banach-Tarski theorem. Restricting to countable unions is exactly the right compromise: powerful enough for limits and infinite series, restrictive enough to remain consistent.
The σ-algebra generated by a collection ℰ, written σ(ℰ), is the smallest σ-algebra containing ℰ — the intersection of all σ-algebras containing ℰ. This intersection is well-defined because the collection of all subsets 𝒫(X) is always a σ-algebra, and any intersection of σ-algebras is again a σ-algebra. The generated σ-algebra is how the Borel σ-algebra on ℝ is built: σ(open sets) = the Borel σ-algebra ℬ(ℝ), which contains all open and closed sets, all countable unions of closed sets (Fσ sets), all countable intersections of open sets (Gδ sets), and iterated combinations thereof.
The formal definition is not arbitrary bookkeeping — it captures exactly which subsets can be measured without contradiction. A function f: X → ℝ is measurable when preimages of measurable sets are measurable: f⁻¹(B) ∈ 𝒜 for every B ∈ ℬ(ℝ). This mirrors the topological definition of continuity (preimages of open sets are open), but for the σ-algebra structure. The interplay between σ(ℰ) and the sets generated by it drives everything downstream — the construction of Lebesgue measure, integration theory, and probability spaces all begin with choosing the right σ-algebra.