Sigma-Algebras and Measurable Sets
A translation-invariant, countably additive length that gives each interval its usual length cannot be assigned to every subset of the line, so first choose a family of sets closed under complement and countable union.
What a learner can do afterwards
- 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
1 · Read
Riemann sums only ever use lengths of intervals. To give sizes to more sets, call a length translation-invariant if shifting a set keeps its length, and countably additive if the length of a union of a sequence of disjoint sets is the sum of their lengths. No such length giving each interval its usual length exists on every subset of the line. So first choose which sets get a size: a sigma-algebra, which contains the whole space and is closed under complements and countable unions.
Take X = {1, 2, 3, 4}. The family {empty set, {1, 2}, {3, 4}, X} is a sigma-algebra, and the family of all 16 subsets is another. The family {empty set, {1}, X} fails because the complement of {1} is {2, 3, 4}, which is missing.
Closure under countable intersections comes for free. The intersection of A1, A2, A3, and so on is the complement of the union of their complements. Each complement is in the family, so is their countable union, and so is the complement of that union. So a sigma-algebra is closed under countable intersections.
The Borel sigma-algebra is the smallest sigma-algebra containing every open interval. Smallest means the intersection of all sigma-algebras that contain those intervals. That family of sigma-algebras is not empty, because the collection of all subsets qualifies. Every member contains the whole space, and if sets lie in every member, so do their complements and countable unions. The intersection is therefore a sigma-algebra.
Finite unions are not enough, because limits produce countable unions. A family closed under complements and finite unions is an algebra. On the natural numbers, call a set cofinite when its complement is finite. The finite or cofinite sets form an algebra that contains every singleton. The even numbers are the countable union of the singletons {2}, {4}, {6}, and so on, yet they are neither finite nor cofinite, so this algebra is not a sigma-algebra.
A sigma-algebra contains the whole space and stays closed under complements and countable unions, so countable limit operations remain available.
Lesson, questions and flashcards are original text written for this dataset.
Reading notes and questions are original text written for this dataset, and flashcards were drafted from collected fact candidates.
2 · Watch
En anglaisTake it off screen
Where it sits
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.