Indicator Function
Definition
The indicator function of an event $A$ is
$$ \mathbf{1}_A(\omega) = I(A) = \begin{cases} 1 & \text{if } \omega \in A, \\ 0 & \text{if } \omega \notin A. \end{cases} $$Key Property
$$ \mathbb{E}[\mathbf{1}_A] = \mathbb{P}(A). $$This identity is the foundation of Monte Carlo probability estimation: estimate $\mathbb{P}(A)$ by averaging indicator samples.