Definition:Measure with Density

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Definition

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

Let $f: X \to \overline \R_{\ge 0}$ be a positive $\mu$-measurable function.


Then the measure with density $f$ with respect to $\mu$, denoted $f \mu$, is defined by:

$\map {f \mu} E := \ds \int_E f \rd \mu$

where $\ds \int_E f \rd \mu$ is the $\mu$-integral of $f$ over $E$.


Also see

  • Results about measure with density can be found here.


Sources