Definition:Integral of Positive Measurable Function over Measurable Set

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

Let $A \in \Sigma$.

Let $f : X \to \overline \R$ be a positive $\Sigma$-measurable function.

Then the $\mu$-integral of $f$ over $A$ is defined by:

$\ds \int_A f \rd \mu = \int \paren {\chi_A \cdot f} \rd \mu$


$\chi_A$ is the characteristic function of $A$
$\chi_A \cdot f$ is the pointwise product of $\chi_A$ and $f$
the integral sign on the right hand side denotes $\mu$-integration of the function $\chi_A \cdot f$.

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