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
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 9$: Problem $5$
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $10.8$