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.
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
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.
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.
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.
A sigma-algebra is your chosen family of measurable sets, closed under complement and countable union so that limits stay inside it.
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8 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.