Definition:Integral of Positive Measurable Function over Measurable Set
Jump to navigation
Jump to search
Definition
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 $\chi_A \cdot f$.
Also see
Sources
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $2.3$: Integral