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Update Chapter 1.md
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YunxiaoZhai authored Dec 13, 2024
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- **Definition**. A collection of subsets of $\Omega$, denoted by F, is called a $\sigma$-field or $\sigma$-algebra if:
$$
\begin{flalign}
\begin{align}
&\text{(i) }\Omega\in\mathcal{F},\\
&\text{(ii) If }A\in\mathcal{F},\text{ then }A^c\in\mathcal{F},\\
&\text{(iii) If }A_1,A_2,\cdots\in\mathcal{F},\text{ then }\cup_{i=1}^\infty A_i\in\mathcal{F}.\end{flalign}
&\text{(iii) If }A_1,A_2,\cdots\in\mathcal{F},\text{ then }\cup_{i=1}^\infty A_i\in\mathcal{F}.\end{align}
$$

- Elements of $\mathcal F$ are called **events**
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