Definition:Measure with Density

Definition
Let $\left({X, \Sigma, \mu}\right)$ 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:


 * $f \mu \left({E}\right) := \displaystyle \int_E f \, \mathrm d \mu$

where $\displaystyle \int_E f \, \mathrm d \mu$ is the $\mu$-integral of $f$ over $E$.

Also see

 * Measure with Density is Measure