The Integral of an Indicator Function is the Measure of the Set

Let $(X,\mathcal{F},\mu)$ be a measure space, with the conventions for $[0,\infty]$ given there, and let $A\in\mathcal{F}$.

Then the indicator function $\mathbf{1}_A$ is a nonnegative simple function on $(X,\mathcal{F})$, and

$$\int_X\mathbf{1}_A\,d\mu=\mu(A),$$

the integral being that of Lebesgue Integral of a Nonnegative Measurable Function.