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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?

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    $\begingroup$ I would probably write the change-of-variables formula as $\int_X f \circ g\,d\mu = \int_Y f\,d(g_*\mu)$. $\endgroup$ – Nate Eldredge Dec 23 '16 at 3:14
  • $\begingroup$ People sometimes write $\int_X f\mu$ instead of $\int_X f\,d\mu$. This yields a good notation for Radon-Nikodym: $\mu = (\mu/\nu)\nu$. To include a coordinate, it would make sense to write $\int_{x\in X} f(x)\mu(x)$ (but people don’t do that). In your example this would give $\int_{x\in X} (f\circ g)(x)\mu(x) = \int_{y\in g(X)} f(y)\mu[g^{-1}(x)]$, properly representing the pushforward measure $g_*\mu(A) = \mu[g^{-1}(A)]$. (People do write $\int_x f(x)\phi(x)\equiv (f,\phi)$ for a distribution $f$ and test function $\phi$, though, so it’s not like the argument $x$ is always taken literally.) $\endgroup$ – Alex Shpilkin Nov 7 '18 at 14:54

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