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
Definition D318
Generated sigma-algebra
Formulation 0
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 \begin{equation} \sigma \langle \mathcal{S} \rangle : = \bigcap \{ \mathcal{F} : \mathcal{S} \subseteq \mathcal{F} \in \mathfrak{S} \} \end{equation}
Formulation 1
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 \begin{equation} \sigma \langle \mathcal{S} \rangle : = \bigcap \{ \mathcal{F} : \mathcal{S} \subseteq \mathcal{F} \text{ and } \mathcal{F} \text{ is a sigma-algebra on } X \} \end{equation}
Children
Borel sigma-algebra
Tail sigma-algebra