One of the strengths of ordinary multivariable calculus is that you can use notation where functions are expressed pointwise (e.g. $\int_a^b x^2 \, \mathrm{d}x$ rather than merely $\int_a^b f$), and after applying all of the insights of differential geometry, we can make sense of path integrals
$$ \int_\gamma u \, \mathrm{d} v$$
for *any* (sufficiently nice) scalar fields $u,v : M \to \mathbb{R}$ and curve $[0,1] \to M$, as well as $\mathrm{d}v$ as being an object in its own right. Part of the key to why this notation works, in my opinion, is that both functions and differential forms *pull back* with respect to smooth maps of manifolds.

Integrals with respect to measures are often notated similarly; e.g. the integral of a measurable $f$ with respect to a measure $\mu$ over a set $E$ is $\int_E f \, \mathrm{d}\mu$. This can be generalized to have a dummy variable; e.g. $\int_E x^2 \, \mathrm{d}\mu(x)$.

However, measures *push forward* with respect to maps of measure spaces; there doesn't seem to be any hope for a notation for integrals with respect to measures where $\mathrm{d}\mu(x)$ behaves like $\mathrm{d}x$ with respect to the sorts of manipulations we use in integral calculus (e.g. $\mathrm{d}(x^2)$ makes sense, but $\mathrm{d}\mu(x^2)$ doesn't seem reasonable). In fact, the part that the measure most resembles is the *region of integration*; e.g. a notation like
$$ \int_{(X,\mu)} f $$
is better behaved; e.g. given any measurable $g:X \to Y$, the change of variable formula would become
$$ \int_{(X,\mu)} f \circ g = \int_{(Y, g_*(\mu))} f $$

This notation doesn't seem particularly convenient either. Thus, my question is

Is there a notation for doing multivariable calculus with measures that has the similar ease-of-use characteristics as traditional multivariable calculus?