--- title: "Sigma-Algebras and Measurable Sets" description: "A size cannot be assigned to every subset of the line, so the first move is to fix the collection of sets the size will be defined on. A sigma-algebra is closed under complement and countable union, w" canonical: https://lightmysky.com/learn/mathematics/sigma-algebras-and-measurable-sets-mt_cpvegazufk source: https://lightmysky.com/learn/mathematics/sigma-algebras-and-measurable-sets-mt_cpvegazufk.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Sigma-Algebras and Measurable Sets A size cannot be assigned to every subset of the line, so the first move is to fix the collection of sets the size will be defined on. A sigma-algebra is closed under complement and countable union, which is exactly what limit arguments later need. Subject: Mathematics · Area: Calculus & Analysis · Ages 22 to 23 Page: https://lightmysky.com/learn/mathematics/sigma-algebras-and-measurable-sets-mt_cpvegazufk ## Ready when they can - Check whether a given collection of sets is a sigma-algebra - Describe the Borel sigma-algebra by what generates it, and say why generation needs an argument - Say why closure under countable unions, rather than finite unions, is the property that matters ## Lesson: Which sets get a size You cannot give a sensible size to every subset of the line. So you first fix the family of sets that will get a size. That family is called a sigma-algebra. It always contains the whole space, and it is closed under complements and countable unions. **Example.** Take X = {1, 2}. The family with only the empty set and X is a sigma-algebra, and so is the family of all four subsets. But the family with the empty set, {1}, and X fails, because the complement of {1} is {2}, which is missing. To test a family, check complements before unions. The Borel sigma-algebra is the smallest sigma-algebra that contains every open interval. Smallest means the intersection of all sigma-algebras that contain those intervals. That step needs a proof, because you must check that intersecting any number of sigma-algebras still leaves a sigma-algebra. **Tip.** Finite unions are not enough, because limits produce countable unions. A set built by a limit, like the points that belong to infinitely many of your sets, needs countably many operations. From complements and countable unions you also get countable intersections, by the De Morgan laws. **Recap.** A sigma-algebra is your chosen family of measurable sets, closed under complement and countable union so that limits stay inside it. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [The Riemann Integral and When a Function Is Integrable](https://lightmysky.com/learn/mathematics/the-riemann-integral-and-when-a-function-is-integrable-mt_Ahl2MYIA4N) - [Countable and Uncountable Sets](https://lightmysky.com/learn/mathematics/countable-and-uncountable-sets-mt_VTHPloNBbJ) ## Opens up - [Outer Measure and the Construction of Lebesgue Measure](https://lightmysky.com/learn/mathematics/outer-measure-and-the-construction-of-lebesgue-measure-mt_6u5_Um71wE) - [Definitions as Design Decisions](https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d) - [Conditional Expectation Given a Sigma-Algebra](https://lightmysky.com/learn/mathematics/conditional-expectation-given-a-sigma-algebra-mt_i2sxZc-ER8)