Sets and σ-fields

Sets

$$\mathcal{P}(\Omega) = 2^\Omega = \{A : A \subseteq \Omega\}$$ $$AB = A \cap B, \qquad A \triangle B = (A \setminus B) \cup (B \setminus A)$$ $$\mathbf{1}_A(\omega) = \begin{cases} 1, & \omega \in A \\ 0, & \omega \notin A \end{cases}$$

Limits of Sets

The inner and outer tails of a sequence $A_1, A_2, \ldots$ are

$$\inf_{k \ge n} A_k = \bigcap_{k=n}^{\infty} A_k, \qquad \sup_{k \ge n} A_k = \bigcup_{k=n}^{\infty} A_k$$ $$\liminf_{n \to \infty} A_n = \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} A_k = \lim_{n \to \infty} \inf_{k \ge n} A_k$$

This is the set of $\omega$ in some inner tail, so $\{n : \omega \notin A_n\}$ is finite. After some $n$, $\omega$ is in all $A_k$, i.e. $\omega$ belongs to all but finitely many $A_k$.

$$\limsup_{n \to \infty} A_n = \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} A_k = \lim_{n \to \infty} \sup_{k \ge n} A_k$$

This is the set of $\omega$ in all outer tails, so $\{n : \omega \in A_n\}$ is infinite. It is the set of outcomes in infinitely many $A_n$, written $\{A_n \text{ i.o.}\}$. Equivalently,

$$\limsup_{n \to \infty} A_n = \Big\{\omega : \sum_{j=1}^{\infty} \mathbf{1}_{A_j}(\omega) = \infty\Big\}$$

Always $\liminf A_n \subseteq \limsup A_n$, and $\lim_{n \to \infty} A_n$ exists if both agree.

For monotone sequences the limit always exists. If $A_1 \subseteq A_2 \subseteq \cdots$ then $\lim A_n = \bigcup_{n} A_n$, written $A_n \uparrow A$. If $A_1 \supseteq A_2 \supseteq \cdots$ then $\lim A_n = \bigcap_{n} A_n$, written $A_n \downarrow A$.

Classes of Sets

• Field Nonempty, closed under complements, finite unions and finite intersections.
• $\sigma$-field Nonempty, closed under complements, countable unions and countable intersections. Equivalently $\emptyset \in \mathcal{C}$, $A \in \mathcal{C} \Rightarrow A^c \in \mathcal{C}$, and $A_1, A_2, \ldots \in \mathcal{C} \Rightarrow \bigcup_{n=1}^{\infty} A_n \in \mathcal{C}$.
• $\pi$-system Nonempty, closed under finite intersections.
• $\lambda$-system Contains $\Omega$, closed under complements and countable disjoint unions.
• Semialgebra A $\pi$-system containing $\emptyset, \Omega$ such that for each $A \in \mathcal{S}$ there are pairwise disjoint $C_1, \ldots, C_n \in \mathcal{S}$ with $A^c = \bigcup_{k=1}^{n} C_k$.

Generated $\sigma$-field

An intersection of $\sigma$-fields is a $\sigma$-field. For $\mathcal{C} \subseteq \mathcal{P}(\Omega)$, let

$$\mathcal{H} = \{\mathcal{B} : \mathcal{B} \text{ is a } \sigma\text{-field on } \Omega,\ \mathcal{C} \subseteq \mathcal{B}\}, \qquad \sigma(\mathcal{C}) = \bigcap_{\mathcal{B} \in \mathcal{H}} \mathcal{B}$$

$\sigma(\mathcal{C})$ is the smallest $\sigma$-field containing every set in $\mathcal{C}$.

The Borel $\sigma$-field on $\mathbb{R}$ is

$$\mathcal{B}(\mathbb{R}) = \sigma(\{(a,b) : -\infty < a < b < \infty\})$$