$\textbf{$\sigma$-Algebra}$
Let $X$ be a set. A collection $\mathscr{A}$ of subsets of $X$ is a $\sigma$-algebra on $X$ if
1) $X\in\mathscr{A}$
2) $A\in\mathscr{A}\implies A^\complement\in \mathscr{A}$
3) $\{A_i\}$ is an infinite sequence in $\mathscr{A}$ $\implies$ $\cup_{i=1}^\infty{A_i}$ belongs to $\mathscr{A}$
Note: It follows from these three conditions that
a) $\varnothing\in \mathscr{A}$
b) $\{A_i\}$ is an infinite sequence in $\mathscr{A}$ $\implies$ $\cap_{i=1}^\infty{A_i}$ belongs to $\mathscr{A}$
$\textbf{$\sigma$-Algebra}$
Let $X$ be a set. A collection $\mathscr{A}$ of subsets of $X$ is a $\sigma$-algebra on $X$ if
1) $X\in\mathscr{A}$
2) $A\in\mathscr{A}\implies A^\complement\in \mathscr{A}$
3) $\{A_i\}$ is an infinite sequence in $\mathscr{A}$ $\implies$ $\cup_{i=1}^\infty{A_i}$ belongs to $\mathscr{A}$
Note: It follows from these three conditions that
a) $\varnothing\in \mathscr{A}$
b) $\{A_i\}$ is an infinite sequence in $\mathscr{A}$ $\implies$ $\cap_{i=1}^\infty{A_i}$ belongs to $\mathscr{A}$
copied
๐-๐๐ฅ๐ ๐๐๐ซ๐
Let ๐ be a set. A collection ๐ of subsets of ๐ is a ๐-algebra on ๐ if
1) ๐โ๐
2) ๐ดโ๐ โน ๐ดแถโ๐
3) {๐ดแตข} is an infinite sequence in ๐ โน โชแตขโโโแชฒ ๐ดแตข belongs to ๐
Note: It follows from these three conditions that
a) โ โ๐
b) {๐ดแตข} is an infinite sequence in ๐ โน โฉแตขโโโแชฒ ๐ดแตข belongs to ๐