ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4328 on D5500: Almost sure event
Probability of finite union with an almost sure event
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E_1, \ldots, E_N \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) \begin{equation} \mathbb{P}(E_1) = 1 \end{equation}
Then \begin{equation} \mathbb{P} \left( \bigcup_{n = 1}^N E_n \right) = 1 \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E_1, \ldots, E_N \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) \begin{equation} \mathbb{P}(E_1) = 1 \end{equation}
This result is a particular case of R4327: Probability of countable union with an almost sure event. $\square$