ThmDex – An index of mathematical definitions, results, and conjectures.
Result R981 on D84: Sigma-algebra
Countably infinite measurable cover has measurable subcover
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 $M$
Let $F_0, F_1, F_2, \ldots$ each be a D11: Set such that
(i) \begin{equation} F_0 : = E_0 \end{equation}
(ii) \begin{equation} \forall \, N \in 1, 2, 3, \ldots : F_N : = E_N \setminus \bigcup_{n = 0}^{N - 1} E_n \end{equation}
Then
(1) \begin{equation} F_0, F_1, F_2, \ldots \in \mathcal{F} \end{equation}
(2) \begin{equation} \forall \, n \in \mathbb{N}: F_n \subseteq E_n \end{equation}
(3) \begin{equation} \bigcup_{n \in \mathbb{N}} E_n = \bigcup_{n \in \mathbb{N}} F_n \end{equation}
(4) \begin{equation} \forall \, n, m \in \mathbb{N} \, (n \geq m \quad \implies \quad F_n \cap F_m = \emptyset) \end{equation}