ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2092 on D198: Probability measure
Sequential continuity of probability measure from above
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) \begin{equation} E_0 \supseteq E_1 \supseteq E_2 \supseteq \cdots \end{equation}
Then \begin{equation} \lim_{n \to \infty} \mathbb{P}(E_n) = \mathbb{P} \left( \bigcap_{n \in \mathbb{N}} E_n \right) \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) \begin{equation} E_0 \supseteq E_1 \supseteq E_2 \supseteq \cdots \end{equation}
Then \begin{equation} \lim_{n \to \infty} \mathbb{P}(E_n) = \mathbb{P} \left( \lim_{n \to \infty} E_n \right) \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) \begin{equation} E_0 \supseteq E_1 \supseteq E_2 \supseteq \cdots \end{equation}
This result is a particular case of R983: Sequential continuity of measure from above. $\square$