Let $(X,\mathfrak{B},\mu)$ be a measure space and $A \in \mathfrak{B}$ .

Let $s \colon X \to [0,\infty]$ be a simple function. Then $\displaystyle \int_A s \, d\mu$ is defined as $\displaystyle \int_A s \, d\mu := \int_X \chi_As \, d\mu$ , where $\chi_A$ denotes the characteristic function of $A$ .

Let $f \colon X \to [0,\infty]$ be a measurable function and
$S=\{s \colon X \to [0,\infty]~~|~~s { is a simple function and } s \le f\}$ . Then $\displaystyle \int_A f \, d\mu$ is defined as $\displaystyle \int_A f \, d\mu := \sup_{s \in S} \int_A s \, d\mu$ .

By the properties of the Lebesgue integral of nonnegative measurable functions (property 3), we have that $\displaystyle \int_A f \, d\mu=\int_X \chi_A f \, d\mu$ .

Let $f \colon X \to [-\infty, \infty]$ be a measurable function such that not both of $\displaystyle \int_A f^+ \, d\mu$ and $\displaystyle \int_A f^- \, d\mu$ are infinite. (Note that $f^+$ and $f^-$ are defined in the entry Lebesgue integral.) Then $\displaystyle \int_A f \, d\mu$ is defined as $\displaystyle \int_A f \, d\mu := \int_A f^+ \, d\mu -\int_A f^- \, d\mu$ .

By the properties of the Lebesgue integral of Lebesgue integrable functions (property 3), we have that $\displaystyle \int_A f \, d\mu=\int_X \chi_A f \, d\mu$ .