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) If $A \in \mathcal A$ then $A^c \in \mathcal A$. 2) If $A_1, A_2 \in \mathcal A$ then $A_1 \cup A_2 \in \mathcal A$.

Sometimes the term "Field of Sets" is used instead of "Algebra of Sets".

Let $X = \{ a, b, c, d, e \}$ and let:

(1)

\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 $X$.

Proposition 1: Let $X$ be a set and let $\mathcal A$ be an algebra on $X$. If $A_1, A_2 \in \mathcal A$ then $A_1 \cap A_2 \in \mathcal A$.

• Proof: Let $A_1, A_2 \in \mathcal A$. Since $\mathcal A$ is an algebra on $X$ we have that $A_1^c, A_2^c \in \mathcal A$ by property (1), and so $A_1^2 \cup A_2^c \in \mathcal A$ by property (2). Observe that:

(2)

\begin{align} \quad A_1^c \cup A_2^c = (A_1 \cap A_2)^c \in \mathcal A \end{align}

• So by property (1) again, $A_1 \cap A_2 \in \mathcal A$. $\blacksquare$

Note: Some people define an algebra on a set $X$ to be a nonempty collection $\mathcal A$ of subsets of $X$ which satisfy (1), (2), and proposition 1 above. Both definitions are equivalent.

Proposition 2: Let $X$ be a set and let $\mathcal A$ be an algebra on $X$. Then $\mathcal A$ contains both $\emptyset$ and $X$.

• Proof: Since $\mathcal A$ is an algebra on $X$, $\mathcal A$ is nonempty. Let $A \in \mathcal A$. By property (1), $A^c \in \mathcal A$. By property (2), $A \cup A^c \in \mathcal A$. But $A \cup A^c = X$. So $X \in \mathcal A$. Also, $X^c = \emptyset \in \mathcal A$ by property (1). $\blacksquare$

If $X$ is a nonempty set then there are always at least two algebras on $X$ called the trivial algebras on $X$.

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