> [!NOTE] Lemma > Let $(\Omega, \mathcal{F}, \mathbb{R})$ be a [[Probability Space|probability space]]. Then the [[Probability Measure|probability measure]] $\mathbb{P}$ satisfies $\mathbb{P}(\emptyset)=0.$ **Proof**: Follows from [[Probability of Subset of Event]]: $\mathbb{P}(\emptyset)=\mathbb{P}(\Omega-\Omega)=\mathbb{P}(\Omega)-\mathbb{P}(\Omega)=1-1=0.$ Proof: Follows from [[Probability of Complement of Event]]: $\mathbb{P}(\emptyset)=\mathbb{P}(\Omega^{c})=1-\mathbb{P}(\Omega)=1-1=0.$