Integral of Characteristic Function/Corollary

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $E \in \Sigma$ be a measurable set, and let $\chi_E: X \to \R$ be its characteristic function.

Then:
 * $\ds \int \chi_E \rd \mu = \map \mu E$

where the integral sign denotes the $\mu$-integral of $\chi_E$.

Proof
By Integral of Characteristic Function, have:


 * $\map {I_\mu} {\chi_E} = \map \mu E$

where $\map {I_\mu} {\chi_E}$ is the $\mu$-integral of $\chi_E$.

From Integral of Positive Measurable Function Extends Integral of Positive Simple Function, it also holds that:


 * $\ds \int \chi_E \rd \mu = \map {I_\mu} {\chi_E}$

Combining these equalities gives the result.