ThmDex – An index of mathematical definitions, results, and conjectures.

Result R1030 on D84: Sigma-algebra

Sigma-algebra is closed under countable intersections

Formulation 1

Let $X$ be a D11: Set such that

(i)	$\mathcal{F}$ is a D84: Sigma-algebra on $X$
(ii)	\begin{equation} E_0, E_1, E_2, \ldots \in \mathcal{F} \end{equation}

Then \begin{equation} \bigcap_{n \in \mathbb{N}} E_n \in \mathcal{F} \end{equation}

Formulation 2

Let $M = (X, \mathcal{F})$ be a D1108: Measurable space such that

(i)	$E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each a D1109: Measurable set in $M$

Then \begin{equation} \bigcap_{n \in \mathbb{N}} E_n \in \mathcal{F} \end{equation}