Algebras of Sets Review
Algebras of Sets Review
We will now review some of the recent material regarding algebras of sets.
- Recall from the Algebras of Sets page that if $X$ is a set then an Algebra on $X$ is a nonempty collection $\mathcal A$ of subsets of $X$ such that for all $A_1, A_2 \in \mathcal A$ we have that $A_1 \cup A_2 \in \mathcal A$ (closure under finite unions), and for all $A \in \mathcal A$ we have that $A^c \in \mathcal A$ (closure under complementation).
- We noted that for any set $X$, $\{ \emptyset, X \}$ and $\mathcal P(X)$ are algebras on $X$ called the Trivial Algebras on $X$.
- On the Properties of Algebras of Sets 1 and Properties of Algebras of Sets 2 page we went through some properties of algebras of sets. These results are summarized in the table below. Let $X$ be a set and let $\mathcal A$ be an algebra on $X$.
| Property 1: | $X \in \mathcal A$ and $\emptyset \in \mathcal A$. |
|---|---|
| Property 2: | If $A_1, A_2 \in \mathcal A$ then $A_1 \cap A_2 \in \mathcal A$. |
| Property 3: | If $A_1, A_2, ..., A_n \in \mathcal A$ then $\displaystyle{\bigcup_{i=1}^{n} A_i \in \mathcal A}$ and $\displaystyle{\bigcap_{i=1}^{n} A_i \in \mathcal A}$. |
| Property 4: | If $A_1, A_2 \in \mathcal A$ then $A_1 \setminus A_2 \in \mathcal A$. |
| Property 5: | If $A, A_1, A_2, ..., A_n \in \mathcal A$ then $\displaystyle{A \setminus \left ( \bigcup_{i=1}^{n} A_i \right ) \in \mathcal A}$. |
| Property 6: | If $(A_n)_{n=1}^{\infty}$ is a sequence of sets in $\mathcal A$ then there exists a sequence of sets $(B_n)_{n=1}^{\infty}$ in $\mathcal A$ such that: (1) the sets in $(B_n)_{n=1}^{\infty}$ are mutually disjoint, and, (2) $\displaystyle{\bigcup_{n=1}^{\infty} A_n = \bigcup_{n=1}^{\infty} B_n}$. |
- We then looked at other algebras. On the Smallest Algebras of a Set X Containing Particular Subsets of X page we say that if $X$ is a set and $\mathcal M$ is a collection of algebras on $X$ then their intersection $\displaystyle{\bigcap_{\mathcal A \in \mathcal M} \mathcal A}$ is an algebra on $X$.
- We used this to show that given any collection $\mathcal C$ of subsets of $X$, there exists a smallest algebra $\mathcal M$ on $X$ containing $\mathcal C$ ($\mathcal C \subseteq \mathcal M$), i.e., if $\mathcal B$ is an algebra on $X$ containing $\mathcal C$ then $\mathcal M \subseteq \mathcal B$.
- On the σ-Algebras of Sets we define another type of algebra. We said that a collection $\mathcal A$ of subsets of $X$ is a $\sigma$-Algebra on $X$ if for all sequences $(A_n)_{n=1}^{\infty}$ of sets in $\mathcal A$ we have that $\displaystyle{\bigcup_{n=1}^{\infty} A_n \in \mathcal A}$ (closure under countable unions) and if for all $A \in \mathcal A$ we have that $A^c \in \mathcal A$ (closure under complementation).
- We noted that every $\sigma$-algebra is itself an algebra. So every result mentioned above also holds for $\sigma$-algebras.
- On the G𝛿 and Fσ Sets page we defined two special types of sets. We define a set $G$ to be a $G_{\delta}$-set if $G$ is a countable intersection of open sets.
- Similarly, we defined a set $F$ to be an $F_{\sigma}$-set if $F$ is a countable union of closed sets.
- We noted that $G$ is a $G_{\delta}$ set if and only if $G^c$ is an $F_{\sigma}$-set.
