ThmDex – An index of mathematical definitions, results, and conjectures.
Result R983 on D1158: Measure space
Sequential continuity of measure from above
Formulation 1
Let $M = (X, \mathcal{F}, \mathbb{\mu})$ be a D1158: Measure space such that
(i) $E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each a D1109: Measurable set in $M$
(ii) \begin{equation} E_0 \supseteq E_1 \supseteq E_2 \supseteq \cdots \end{equation}
(iii) \begin{equation} \exists \, n \in \mathbb{N} : \mu(E_n) < \infty \end{equation}
Then \begin{equation} \lim_{n \to \infty} \mu(E_n) = \mu \left( \bigcap_{n \in \mathbb{N}} E_n \right) \end{equation}
Formulation 3
Let $M = (X, \mathcal{F}, \mathbb{\mu})$ be a D1158: Measure space such that
(i) $E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each a D1109: Measurable set in $M$
(ii) \begin{equation} E_0 \supseteq E_1 \supseteq E_2 \supseteq \cdots \end{equation}
(iii) \begin{equation} \exists \, n \in \mathbb{N} : \mu(E_n) < \infty \end{equation}
Then \begin{equation} \lim_{n \to \infty} \mu(E_n) = \mu \left( \lim_{n \to \infty} E_n \right) \end{equation}