Sigma-algebras and measures
Objective. Construct measurable spaces and verify measures.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A σ-algebra on a set X is a collection of subsets closed under complements and countable unions, containing X itself, and a measure is a countably additive function on that collection taking values in [0,∞]. The three axioms — ∅ and X in the collection, closure under complementation, and closure under countable unions — guarantee that familiar operations like countable intersections also stay inside the collection via De Morgan's laws. Measures must satisfy μ(∅)=0 and countable additivity: μ(∪Aₙ)=Σμ(Aₙ) for disjoint Aₙ. For example, on ℝ the Borel σ-algebra is generated by open intervals, and Lebesgue measure assigns length ℓ([a,b])=b−a to intervals and extends countably additively to all Borel sets.
Worked example
Let 𝒜 be the σ-algebra generated by {A,B} on X={1,2,3,4}, A={1,2}, B={2,3}. Find |𝒜|.
- 1Start from atoms: A∩B={2}, A∖B={1}, B∖A={3}, rest={4}
- 2These four disjoint atoms generate all unions in 𝒜
- 3Number of subsets of 4 atoms is 2⁴
- 4So |𝒜| = 16
Answer: 16
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Practice
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Start practisingCommon mistake. Students think any union of measurable sets is automatically measurable without checking closure is only guaranteed for countable unions; an uncountable union of singletons need not be measurable in general σ-algebras. Always confirm the index set is countable before applying the axiom.
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