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 $\caps{}{n\in\N}E_{n}=X\del \cups{}{n\in\N}(X\del E_{n})$
$\square$
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