Measure Theory/Basic Structures And Definitions/Semialgebras, Algebras and σ-algebras

Semialgebras

Roughly speaking, a semialgebra over a set $X$ is a class that is closed under intersection and semi closed under set difference. Since these restrictions are strong, it's very common that the sets in it have a defined characterization and then it's easier to construct measures over those sets. Then, we'll see the structure of an algebra, that it's closed under set difference, and then the σ-algebra, that it is an algebra and closed under countable unions. The first structures are of importance because they appear naturally on sets of interest, and the last one because it's the central structure to work with measures, because of it's properties.

Algebras

An algebra over a set $X$ it's a class closed under all finite set operations

This definition suffices for the closure under finite operations. The following properties shows it

Template:MT book:Proposition

Note: It's easy to see that $\mathrm{\forall }A,B\in 𝒜\Rightarrow A\cup B\in 𝒜$ and then an algebra it's closed for all finite set operations.

σ-algebras

A σ-algebra over a set $X$ is an algebra closed under countable unions

Note: A σ-algebra it's also closed under countable intersections, because the complement of a countable union, is the countable intersection of the complement of the sets considered in the union.