Category:Integral of Positive Measurable Function over Measurable Set
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This category contains results about Integral of Positive Measurable Function over Measurable Set.
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:
- $\ds \int_A f \rd \mu = \int \paren {\chi_A \cdot f} \rd \mu$
where:
- $\chi_A$ 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 $\chi_A \cdot f$.
Pages in category "Integral of Positive Measurable Function over Measurable Set"
The following 5 pages are in this category, out of 5 total.
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- Integral of Positive Measurable Function is Additive/Corollary
- Integral of Positive Measurable Function is Monotone/Corollary
- Integral of Positive Measurable Function is Positive Homogeneous/Corollary
- Integral of Positive Measurable Function over Disjoint Union
- Integral of Positive Measurable Function over Measurable Set is Well-Defined