> [!NOTE] Theorem > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $A\in \mathcal{F}.$ Let $\mathbb{1}_{A}$ denote the [[Event Indicator Function|indicator function]] for $A.$ Then $\mathbb{1}_{A}$ is a [[Discrete random variables|discrete real-valued random variable]] on $(\Omega, \mathcal{F}, \mathbb{P})$ and its [[Expectation of Discrete Real-Valued Random Variable|expectation]] is given by $\mathbb{E}[\mathbb{1}_{A}]=\mathbb{P}(A)$ (so it is [[Integrable Discrete Real-Valued Random Variable|integrable]]). **Proof**: By [[Indicator Function is Discrete Real-Valued Random Variable]], $\mathbb{1}_{A}$ is indeed a discrete random variable. Its expectation is given by $\mathbb{E}[\mathbb{1}_{A}]=0 \times \mathbb{P}(\Omega\setminus A)+1 \times \mathbb{P}(A)=\mathbb{P}(A).$