Lebesgue integral over a subset of the measure space

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 $\displaystyle\int_{A}s\,d\mu:=\int_{X}\chi_{A}s\,d\mu$ , where $\chi_{A}$ denotes the characteristic function of $A$ .

Let $f\colon X\to[0,\infty]$ be a measurable function and
$S=\{s\colon X\to[0,\infty]~{}~{}|~{}~{}s\text{ is a simple function and }s\leq f\}$ . 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 $\displaystyle\int_{A}f\,d\mu=\int_{X}\chi_{A}f\,d\mu$ .

Title	Lebesgue integral over a subset of the measure space
Canonical name	LebesgueIntegralOverASubsetOfTheMeasureSpace
Date of creation	2013-03-22 16:13:54
Last modified on	2013-03-22 16:13:54
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	7
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 26A42
Classification	msc 28A25