PlanetMath: Lebesgue integral over a subset of the measure space

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Lebesgue integral over a subset of the measure space

(Definition)

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 .

Let $f \colon X \to [0,\infty]$ be a measurable function and
$S=\{s \colon X \to [0,\infty]~~\vert~~s$ 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 and 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 $\displaystyle \int_A f \, d\mu=\int_X \chi_A f \, d\mu$ .

"Lebesgue integral over a subset of the measure space" is owned by Wkbj79.

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Cross-references: properties of the Lebesgue integral of Lebesgue integrable functions, Lebesgue integral, infinite, properties of the Lebesgue integral of nonnegative measurable functions, measurable function, characteristic function, simple function, measure space
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This is version 4 of Lebesgue integral over a subset of the measure space, born on 2006-09-09, modified 2007-06-29.
Object id is 8332, canonical name is LebesgueIntegralOverASubsetOfTheMeasureSpace.
Accessed 1200 times total.

Classification:

AMS MSC:	28A25 (Measure and integration :: Classical measure theory :: Integration with respect to measures and other set functions)
	26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)

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