ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Subset ▼ Power set ▼ Hyperpower set sequence ▼ Hyperpower set ▼ Hypersubset ▼ Subset algebra ▼ Boolean algebra ▼ Sigma-algebra ▼ Set of sigma-algebras ▼ Generated sigma-algebra
Definition D3889
Tail sigma-algebra

Let $X$ be a D11: Set such that
 (i) $\mathcal{F}_0, \mathcal{F}_1, \mathcal{F}_2, \ldots \subseteq \mathcal{P}(X)$ are each a D84: Sigma-algebra on $X$
The tail sigma-algebra on $X$ with respect to $\mathcal{F}_0, \mathcal{F}_1, \mathcal{F}_2, \dots$ is the D11: Set $$\bigcap_{n \in \mathbb{N}} \sigma \left\langle \bigcup_{m \in \mathbb{N} : m \geq n} \mathcal{F}_m \right\rangle$$

Let $X$ be a D11: Set such that
 (i) $\mathcal{F}_0, \mathcal{F}_1, \mathcal{F}_2, \ldots \subseteq \mathcal{P}(X)$ are each a D84: Sigma-algebra on $X$
The tail sigma-algebra on $X$ with respect to $\mathcal{F}_0, \mathcal{F}_1, \mathcal{F}_2, \dots$ is the D11: Set $$\sigma \left\langle \mathcal{F}_0 \cup \mathcal{F}_1 \cup \cdots \right\rangle \cap \sigma \left\langle \mathcal{F}_1 \cup \mathcal{F}_2 \cup \cdots \right\rangle \cap \sigma \left\langle \mathcal{F}_2 \cup \mathcal{F}_3 \cup \cdots \right\rangle \cdots$$

Let $X$ be a D11: Set such that
 (i) $\mathcal{F}_0, \mathcal{F}_1, \mathcal{F}_2, \ldots \subseteq \mathcal{P}(X)$ are each a D84: Sigma-algebra on $X$
The tail sigma-algebra on $X$ with respect to $\mathcal{F}_0, \mathcal{F}_1, \mathcal{F}_2, \dots$ is the D11: Set $$\sigma \left\langle \bigcup_{m = 0}^{\infty} \mathcal{F}_n \right\rangle \cap \sigma \left\langle \bigcup_{m = 1}^{\infty} \mathcal{F}_n \right\rangle \cap \sigma \left\langle \bigcup_{m = 2}^{\infty} \mathcal{F}_n \right\rangle \cdots$$