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Definition D318
Generated sigma-algebra

Let $X$ be a D11: Set such that
 (i) $\mathfrak{S} : = \mathfrak{S}(X)$ is the D484: Set of sigma-algebras on $X$ (ii) $\mathcal{S} \subseteq \mathcal{P}(X)$ is a D3367: Subset algebra on $X$
The generated sigma-algebra on $X$ with generator $\mathcal{S}$ is the D11: Set $$\sigma \langle \mathcal{S} \rangle : = \bigcap \{ \mathcal{F} : \mathcal{S} \subseteq \mathcal{F} \in \mathfrak{S} \}$$

Let $X$ be a D11: Set such that
 (i) $\mathcal{S} \subseteq \mathcal{P}(X)$ is a D3367: Subset algebra on $X$
The generated sigma-algebra on $X$ with generator $\mathcal{S}$ is the D11: Set $$\sigma \langle \mathcal{S} \rangle : = \bigcap \{ \mathcal{F} : \mathcal{S} \subseteq \mathcal{F} \text{ and } \mathcal{F} \text{ is a sigma-algebra on } X \}$$
Children
 ▶ Borel sigma-algebra ▶ Tail sigma-algebra