Measure with Density is Measure

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Theorem

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 $f \mu$, the measure with density $f$ with respect to $\mu$ is a measure.


Proof


Sources