4.1 Algebras
Definition 4.1.1 (Algebra).label Let $X$ be a set and $\Si\suf 2^{X}$ then $\Si$ is an algebra if
- (A1)
$\emp,X\in\Si$
- (A2)
$\fall E\in \Si,X\del E\in\Si$
- (A3)
$\fall E,F\in\Si,E\cup F\in\Si$
Definition 4.1.2 (Ring).label Let $X$ be a set and $\cali{A}\suf 2^{X}$, then $\cali{A}$ is a ring if:
- (A1)
$\emp,X\in\cali{A}$
- (A2’)
$\fall E,F\in\cali{A}$ with $E\suf F,F\del E\in\cali{A}$
- (A3)
$\fall E,F\in\cali{A},E\cup F\in\cali{A}$
Definition 4.1.3 ($\s$-Algebra).label Let $X$ be a set then a collection $\Si\suf 2^{X}$ is a $\s$-algebra of subsets of $X$ if
- (A1)
$\emp,X\in\Si$
- (A2)
$\fall E\in \Si,X\del E\in\Si$
- (A3)
$\fall \curl{E_n}_{n\in\N}\suf\Si,\cups{}{n\in\N}E_{n}\in\Si$
If $\Si$ is a $\s$-algebra then
For each $E,F\in\Si$ we have $E\cup F,E\cap F,E\del F\in \Si$
For each $\curl{E_n}_{n\in\N}\suf\Si$ we have $\caps{}{n\in\N}E_{n}\in \Si$
Proof.
$E\cup F\in\Si$ by taking the sequence $E_{0}=E,E_{n}=F$ for all $n\geq 1$.
Since $X\del E,X\del F\in\Si$ we have $E\cap F=X\del ((X\del E)\cup (X\del F))\in \Si$.
Lastly $E\del F=E\cap (X\del F)\in\Si$.
Observe $\caps{}{n\in\N}E_{n}=X\del \cups{}{n\in\N}(X\del E_{n})$
$\square$
Lemma 4.1.4.label Let $X$ be a set and $\cali{A}\suf 2^{X}$ be an algebra, then the following are equivalent:
- (1)
For any $\curl{A_n }_{n\in\N}\suf \cali{A},\cups{}{n\in\N}A_{n}\in\cali{A}$.
- (2)
For any $\curl{A_n }_{n\in\N}\suf \cali{A}$ with $A_{n}\suf A_{n+1}$ for all $n\in\N,\cups{}{n\in\N}A_{n}\in\cali{A}$.
- (3)
For any $\curl{A_n }_{n\in\N}\suf \cali{A}$ pairwise disjoint, $\dcups{}{n\in\N}A_{n}\in\cali{A}$.
Proof. (2)$\implies$(1): For each $n\in\N$ let $B_{n}=\cups{}{k\leq n}A_{k}\in\cali{A}$ then $\cups{}{n\in\N}A_{n}=\cups{}{n\in\N}B_{n}\in\cali{A}$.
(3)$\implies$(2): Let $A_{0}=\emp$ and $B_{n}=A_{n}\del A_{n-1}$ for all $n\in\N$ then $\curl{B_n}_{n\in\N}\suf\cali{A}$ is pairwise disjoint and $\cups{}{n\in\N}A_{n}=\dcups{}{n\in\N}B_{n}\in\cali{A}$.$\square$
Definition 4.1.5 (Generated $\s$-Algebra).label Let $X$ be a set and $\cali{E}\suf 2^{X}$ then the $\s$-algebra $\s(\cali{E})$ generated by $\cali{E}$ is the smallest $\s$-algebra on $X$ containing $\cali{E}$
Definition 4.1.6 (Induced $\s$-Algebra).label Let $X$ be a set, $\Si\suf 2^{X}$ be a $\s$-algebra over $X$ and $E\suf X$ then the collection
is the $\s$-algebra on $E$ induced by $\Si$
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