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Definition D3889
Tail sigma-algebra
Formulation 0
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 \begin{equation} \bigcap_{n \in \mathbb{N}} \sigma \left\langle \bigcup_{m \in \mathbb{N} : m \geq n} \mathcal{F}_m \right\rangle \end{equation}
Formulation 1
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 \begin{equation} \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 \end{equation}
Formulation 2
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 \begin{equation} \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 \end{equation}