ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4073 on D1159: Probability space
Probability of union with the impossible event
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E \in \mathcal{F}$ is an D1716: Event in $P$
Then \begin{equation} \mathbb{P}(E \cup \emptyset) = \mathbb{P}(E) \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E \in \mathcal{F}$ is an D1716: Event in $P$
Then \begin{equation} \mathbb{P}(E \text{ or } \emptyset) = \mathbb{P}(E) \end{equation}
Formulation 2
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E \in \mathcal{F}$ is an D1716: Event in $P$
Then \begin{equation} \mathbb{P}(E \lor \emptyset) = \mathbb{P}(E) \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E \in \mathcal{F}$ is an D1716: Event in $P$
Result R2082: Binary set union with empty set shows that $E \cup \emptyset = E$, whence the claim follows. $\square$