Measure with a Density

In the setting of Measurable Functions and the Lebesgue Integral: Notation, with its measure space $(X,\mathcal{F},\mu)$, let $h\in\mathcal{M}^{+}(X,\mathcal{F})$.

(i) Measure with density $h$. The measure with density $h$ with respect to $\mu$ is the function $h\mu:\mathcal{F}\to[0,\infty]$ with

$$(h\mu)(A)=\int_{A}h\,d\mu\qquad(A\in\mathcal{F}),$$

the integral of $h$ over $A$.