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

SemialgebrasEdit

Roughly speaking, a semialgebra over a set   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 its properties.

Definition 1.1.1: A class   is a Semialgebra over   if:

  • The empty set and whole set are in  :
 
  • It's closed under intersection:
 
  • The set difference of any two sets in   is the finite disjoint union of elements in  :
  pairwise-disjoint such that  

Example: It might seem—at first sight—that a semialgebra is a very restricted subset of  , but it's easy to prove that with   the class of all intervals (bounded, unbounded, semi-open, open, closed or any other class) is a semialgebra over   and clearly this set is non-trivial. For example, let A be   and B be  . Then  , say. Let us call   and  . Then   (because it is not an interval) even though  . Further,   and   are disjoint.

AlgebrasEdit

An algebra over a set   is a class closed under all finite set operations.

Definition 1.1.2 : A class   is an Algebra over   if:

  1.  
  2.  

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

Proposition 1.1 : A class   is an algebra if and only if   satisfies :

  1.  
  2.  
  3.  

Proof :  

Property 1 is identical.

For property 2 , note that  :

 

Finally for property 3 , since the property 2 holds,   :

 

 

Property 1 is identical.

For all  , from property 2 we have that  . Property 3 then implies that  , which is equivalent to    

Note: It's easy to see that given  , then, from properties 2 and 3,  , so an algebra is closed for all finite set operations.

σ-algebrasEdit

A σ-algebra (also called σ-ring) over a set   is an algebra closed under countable unions.

Definition 1.1.3 : A class   is a σ-algebra over   if:

  1.   is an algebra
  2.  

Note: A σ-algebra is 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.

Borel SetsEdit

TheoremEdit

Let   be a set and let   be a collection of subsets of  . Then, there exists a smallest σ-ring   containing  , that is, if   is a σ-ring containing  , then  

ProofEdit

Let   be the intersection of all σ-rings that contain  . It is easy to see that   and that   and thus,   is a σ-ring.


  is sometimes said to be the extension of  

Now, let   be a topology over  . Thus, there exists a σ-algebra   over   such that  .   is called Borel algebra and the members of   are called Borel sets