Measure-Theoretic Lingo

Let $\Omega$ be a set. A collection $\mathcal F \subseteq \mathcal P(\Omega)$ is a $\sigma$ -algebra on $\Omega$ if it satisfies the following properties:

$\emptyset, \Omega \in \mathcal F$ ,
For any $M \in \mathcal F$ , $\Omega \backslash M \in \mathcal F$ .
For any $M_1,M_2,\dots, \in \mathcal F$ , $\displaystyle \bigcup_{i=1}^\infty X_i \in \mathcal F$ .

If such a $\sigma$ -algebra exists, we call $(\Omega,\mathcal F)$ a measurable space. Elements of $\mathcal F$ are called measurable sets.

Example 1. For any set $\Omega$ , $(\Omega, \{\emptyset, \Omega\})$ is a $\sigma$ -algebra known as the trivial $\sigma$ -algebra) and that $(\Omega, \mathcal P(\Omega))$ is a $\sigma$ -algebra.

Problem 1. Let $\Omega$ be a set and fix $\mathcal K \subseteq \mathcal P(\Omega)$ . Find a $\sigma$ -algebra that contains $\mathcal K$ , then define

$\displaystyle \mathcal K' := \{\mathcal G \subseteq \mathcal P(\Omega) : \mathcal K \subseteq \mathcal G,\ \mathcal G\ \text{is a}\ \sigma\text{-algebra}\},\quad \sigma(\mathcal K) := \bigcap_{\mathcal G \in \mathcal K'}\mathcal G.$

Prove that $\sigma(\mathcal K)$ is the smallest $\sigma$ -algebra that contains $\mathcal K$ , called the $\sigma$ -algebra generated by $\mathcal K$ .

(Click for Solution)

Solution. Obviously, $\mathcal K \subseteq \mathcal P(\Omega)$ . For the first property, since $\emptyset,\Omega \in \mathcal G$ for any $\mathcal G \supseteq \mathcal K$ , $\emptyset, \Omega \in \sigma(\mathcal K)$ .

For the second property, fix $M \in \sigma(\mathcal K)$ . Then $M \in \mathcal G$ for any $\mathcal K \subseteq \mathcal G$ . Since each $\mathcal G$ is a $\sigma$ -algebra, $\Omega \backslash M \in \mathcal G$ . Hence, $\Omega \backslash M \in \sigma(\mathcal K)$ .

For the third property, fix $M_1,M_2,\dots \in \sigma(\mathcal K)$ . Then for each $i$ , $M_i \in \mathcal G$ for any $\mathcal K \subseteq \mathcal G$ . Since each $\mathcal G$ is a $\sigma$ -algebra, $\bigcup_{i=1}^\infty M_i \in \mathcal G$ . Hence, $\bigcup_{i=1}^\infty M_i \in \sigma(\mathcal K)$ .

Problem 2. Let $(\Omega, \mathcal F)$ be a measurable space. For any $K \subseteq \Omega$ , define the collection $\mathcal F_K \subseteq \mathcal P(K)$ of subsets by