Algebras of Sets
Table of Contents

Algebras of Sets

We begin by defining the important concept of an algebra on a set. This will be used more extensively later as we delve into more complicated measure theory topics.

Definition: Let $X$ be a set. An Algebra on $X$ is a nonempty collection $\mathcal A$ of subsets of $X$ with the following properties:
1) For every set $A_1, A_2 \in \mathcal A$ we have that the union, $A_1 \cup A_2 \in \mathcal A$.
2) For every set $A \in \mathcal A$ we have that the complement, $A^c \in A$.

Since $\mathcal A$ by definition is a collection of subsets of $X$ we have that $\mathcal A \subseteq \mathcal P (X)$ where $\mathcal P(X)$ is the set of all subsets of $X$ called the Power Set on $X$.

A collection of subsets $\mathcal A$ of $X$ is an algebra if the union of any two sets in $\mathcal A$ is also in $\mathcal A$ (that is, $\mathcal A$ is closed under the union of two sets), and if for any set in $\mathcal A$ we have that the complement of that set is also in $\mathcal A$ (that is, $\mathcal A$ is closed under complements).

The simplest algebra on a set $X$ is:

(1)
\begin{align} \quad \mathcal A = \{ \emptyset, X \} \end{align}

Clearly property (1) holds since $\emptyset \cup X = X \in \mathcal A$. Furthermore, property (2) holds since $\emptyset^c = X \in \mathcal A$ and $X^c = \emptyset \in \mathcal A$.

Another simple algebra on a set $X$ is:

(2)
\begin{align} \quad \mathcal A = \mathcal P(X) \end{align}

It is clear that properties (1) and (2) hold as any union of subsets of $X$ is also a subset of $X$ (and hence in $\mathcal P(X)$), and the complement of any subset of $X$ is a subset of $X$ (and once against is in $\mathcal P(X)$).

The two algebras on a generic set $X$ described above are sometimes called the trivial algebras on $X$. We define this below for completeness.

Definition: Let $X$ be a set. The Trivial Algebras on $X$ are the algebras $\{ \emptyset, X \}$ and $\mathcal P (X)$.

Let's look at a nontrivial example. Let $X = \{ a, b, c, d, e \}$ and let:

(3)
\begin{align} \quad \mathcal A = \{ \emptyset, \{ a, b \}, \{ c, d, e \}, X \} \end{align}

It is easy to verify that properties (1) and (2) from the definition of an algebra on a set hold. Thus $\mathcal A$ is an algebra on $X$.

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