> [!NOTE] Definition > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $A\in \mathcal{F}.$ Define $\mathbb{1}_{A}:\Omega\to \{ 0,1 \}$ by $\mathbb{1}_{A}(\omega)=\begin{cases}1, & \omega\in A \\ 0, &\omega \in \Omega \setminus A .\end{cases}$ # Properties By [[Indicator Function is Discrete Real-Valued Random Variable]], $\mathbb{1}_{A}$ is indeed a discrete real valued random variable on $(\Omega, \mathcal{F}, \mathbb{P}).$ By [[Expectation of Discrete Real-Valued Random Variable is Unitary]], $\mathbb{E}[\mathbb{1}_{A}]=\mathbb{P}(A).$