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 its properties.

Definition 1.1.1 : A class $\mathcal{S} \subseteq \mathcal{P}(X)$ is a Semialgebra over $\,X$ if :

• The empty set and whole set are in $\mathcal{S}$:

$\emptyset \in \mathcal{S}, X \in \mathcal{S}$

• Its closed under intersection :

$\forall A, B \in S \Rightarrow A \cap B \in \mathcal{S}$

• The set difference of any two sets in $\mathcal{S}$ is the finite union of elements in $\mathcal{S}$ :

$\forall A, B \in S, \; \; \exist \{C_i\}_{i=1}^{n} \subseteq \mathcal{S}, \; A \setminus B = \bigcup_{i=1}^{n} C_i$

Example : It might seem—at first sight—that a semialgebra is a very restricted subset of $\mathcal{P}(X)$, but it's easy to prove that with $X = \mathbb{R}$ the class :

$\qquad \mathcal{S} = \{(a,b] : a \leq b \in \mathbb{R} \}$

is a semialgebra over $\mathbb{R}$ and clearly this set is non-trivial.

Algebras

An algebra over a set $\,X$ is a class closed under all finite set operations.

Definition 1.1.2 : A class $\mathcal{A} \subseteq \mathcal{P}(X)$ is an Algebra over $\,X$ if:

1. $X \in \mathcal{A}$
2. $\forall A,B \in \mathcal{A} \Rightarrow A \setminus B \in \mathcal{A}$

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

Proposition 1.1 : A class $\mathcal{A} \subseteq \mathcal{P}(X)$ is an algebra if and only if $\mathcal{A}$ satisfies :

1. $X \in \mathcal{A}$
2. $\forall A \in \mathcal{A} \Rightarrow A^c \in \mathcal{A}$
3. $\forall A, B \in \mathcal{A} \Rightarrow A \cap B \in \mathcal{A}$

Proof : $(\Rightarrow)$

Property 1, it's direct.

For property 2 , note that $\forall A \in \mathcal{A}$:

$A^c = X \setminus A \in \mathcal{A}$

Finally for property 3 , since the property 2 holds, $\forall B \in \mathcal{A}$ :

$B^c \in \mathcal{A} \Rightarrow A \setminus B^c = A \cap B \in \mathcal{A}$

$(\Leftarrow)$

Trivially $X \in \mathcal{A}$

and using properties 2 and 3 it's clear that $\forall A,B \in \mathcal{A} \Rightarrow A\setminus B \in \mathcal{A}$ $\Box$

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

σ-algebras

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

Definition 1.1.3 : A class $\mathcal{T} \subseteq \mathcal{P}(X)$ is an σ-algebra over $\,X$ if:

1. $\mathcal{T}$ is an algebra
2. $\forall \{{A_{n}\}}_{n \in \mathbb{N}} \subset \mathcal{T} \Rightarrow \cup_{n \in \mathbb{N}} A_{n} \in \mathcal{T}$

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 Sets

Theorem

Let $X$ be a set and let $\mathcal{F}$ be a collection of subsets of $X$. Then, there exists a smallest σ-ring $\mathcal{F}^*$ containing $\mathcal{F}$, that is, if $\mathcal{M}$ is a σ-ring containing $\mathcal{F}$, then $\mathcal{F}^*\subseteq\mathcal{M}$

Proof

Let $\mathcal{F}^*$ be the intersection of all σ-rings that contain $\mathcal{F}$. It is easy to see that $E\in\mathcal{F}^*\Rightarrow E^c\in\mathcal{F}^*$ and that $\mathcal{M}_1,\mathcal{M}_2,\ldots\in\mathcal{F}^*\Rightarrow\bigcup_{i=1}^{\infty}\mathcal{M}_i\in\mathcal{F}^*$ and thus, $\mathcal{F}^*$ is a σ-ring.

$\mathcal{F}^*$ is sometimes said to be the extension of $\mathcal{F}$

Now, let $\mathcal{T}$ be a topology over $X$. Thus, there exists a σ-algebra $\mathcal{B}$ over $X$ such that $\mathcal{B}=\mathcal{T}^*$. Members of $\mathcal{B}$ are called Borel sets