A sigma algebra (or σ-algebra) is to set algebras what sigma males are to males. The Borel hierarchy (the analog of the male hierarchy) is not a σ-algebra at any countable rank; instead the σ-algebra has the largest possible rank -- ω_1, the least uncountable ordinal (under suitable conditions e.g. the Polish space is uncountable. This is known as the non-collapse of the Borel hierarchy).
A σ-algebra is closed under countable union, countable intersection, and relative complement. This means σ-algebras arise naturally in the study of measure theory, where the measurable sets form a σ-algebra.
A σ-algebra is closed under countable union, countable intersection, and relative complement. This means σ-algebras arise naturally in the study of measure theory, where the measurable sets form a σ-algebra.
The algebra of Lebesgue-measurable subsets of R (i.e. the sets of real numbers with a well-defined "length", which is possibly infinity) form a sigma algebra.
by sigma_algebra_grindset June 5, 2022