Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Power set
Hyperpower set sequence
Hyperpower set
Subset algebra
Boolean algebra
Formulation 0
Let $X$ be a D11: Set such that
(i) $\mathcal{P}(X)$ is the D80: Power set of $X$
A D11: Set $\mathcal{F} \subseteq \mathcal{P}(X)$ is a sigma-algebra on $X$ if and only if
(1) \begin{equation} \emptyset \in \mathcal{F} \end{equation}
(2) \begin{equation} \forall \, E \in \mathcal{F} : X \setminus E \in \mathcal{F} \end{equation}
(3) \begin{equation} \forall \, E_0, E_1, E_2, \dots \in \mathcal{F} : \bigcup_{n \in \mathbb{N}} E_n \in \mathcal{F} \end{equation}
[Comment 0] : Occasionally, one might see condition (1) requiring that both $\emptyset$ and $X$ are in $\mathcal{F}$. However, results R2067: Subtracting empty set from set and R2066: Difference of set with itself show that $X \setminus \emptyset = X$ and $X \setminus X = \emptyset$, so that condition (2) ensures $\emptyset, X \in \mathcal{F}$ if only one of $\emptyset$ or $X$ is assumed to belong to $\mathcal{F}$.
Also known as
Child definitions
» D1916: Bottom sigma-algebra
» D195: Discrete sigma-algebra
» D1729: Pushforward sigma-algebra
» D484: Set of sigma-algebras
» D470: Subsigma-algebra
» R1030: Sigma-algebra is closed under countable intersections
» R981: Countably infinite measurable cover has measurable subcover
» R4489: Sigma-algebra is closed under limit superiors and limit inferiors
» R2270: Sigma-algebra is closed under symmetric differences
» R4648: Union of sigma-algebras need not be a sigma-algebra