I have a little trouble understanding the notation of the infinitesimal measure in Lebesgue integration. For example, let's assume I want to compute an volume integral of a function $f: D \rightarrow R^3$ ($D \subset R^3$) and let's also assume I don't care about the orientation of the volume. Given a measure space $(D, S, \mu)$ my integral then reads (found on the internet):
$\int\limits_D f(x) \, \mu(\mathrm{d} x)$.
How do I interpret $\mu(\mathrm{d} x)$, i.e. the measure of a vector? Wouldn't it make more sense to use $\mu(\mathrm{d} D)$ and define $\mathrm{d} D$ as the (possibly disjoint) subset of $D$ upon which $f$ takes on the equal values?
Best, John