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

Result R4489 on D84: Sigma-algebra

Sigma-algebra is closed under limit superiors and limit inferiors

Formulation 0

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

(1)	\begin{equation} \bigcap_{n = 1}^{\infty} \bigcup_{m = n}^{\infty} E_m \in \mathcal{F} \end{equation}
(2)	\begin{equation} \bigcup_{n = 1}^{\infty} \bigcap_{m = n}^{\infty} E_m \in \mathcal{F} \end{equation}

Formulation 1

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

(1)	\begin{equation} \limsup_{n \to \infty} E_n \in \mathcal{F} \end{equation}
(2)	\begin{equation} \liminf_{n \to \infty} E_n \in \mathcal{F} \end{equation}