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Update Chapter 1.md
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YunxiaoZhai authored Dec 13, 2024
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### 1.2 $\sigma(\mathcal A)$ & Borel $\sigma$-field

- **Definition**. Let $\mathcal A$ be a collection of subsets of $\Omega$, then define $\sigma$-field as $\sigma(\mathcal A):=\underset{\mathcal A\subset\mathcal F_i,\mathcal F_i\text{ is $\sigma$-field for all }i\in I}\cap \mathcal F_i$.
- **Definition**. Let $\mathcal A$ be a collection of subsets of $\Omega$, then define $\sigma$-field as $\sigma(\mathcal A):=\underset{\mathcal A\subset\mathcal F_i,\mathcal F_i\text{ is} \sigma\text{-field for all }i\in I}\cap \mathcal F_i$.

- Simply put it, $\sigma(\mathcal A)$ is smallest $\sigma$-algebra containing $\mathcal A$
- We also call $\sigma(\mathcal A)$ as $\sigma$-algebra generated by $\mathcal A$
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