ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D84
Sigma-algebra

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) $$\emptyset \in \mathcal{F}$$ (2) $$\forall \, E \in \mathcal{F} : X \setminus E \in \mathcal{F}$$ (3) $$\forall \, E_0, E_1, E_2, \dots \in \mathcal{F} : \bigcup_{n \in \mathbb{N}} E_n \in \mathcal{F}$$
 ▶▶▶ 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 $\mathcal{F}$.
Children
 ▶ D1916: Bottom sigma-algebra ▶ D195: Discrete sigma-algebra ▶ D1729: Pushforward sigma-algebra ▶ D484: Set of sigma-algebras ▶ D470: Subsigma-algebra