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The integral of a function of $n$ variables $f(x_1,x_2,\dots,x_n)$ can be expressed as:

$$\int \cdots \int_\mathbf{D}\, f(x_1,x_2,\ldots,x_n) \,dx_1 \!\cdots dx_n$$

However this is a rather cumbersome notation and I am wondering whether it is possible to instead reparameterise the functions as $f(\boldsymbol{\chi})$ where $\boldsymbol{\chi} = \{x_1,x_2,\dots,x_n\}$ and then express the integral as:

$$\int_\mathbf{D} f(\boldsymbol{\chi})\,d\boldsymbol{\chi}$$

Is this more succint notation possible or is it too misleading?

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  • $\begingroup$ One common notation is $\int_D f \, dx$. This notation is used in Wheeden and Zygmund, for example. You can also use $\int_D f(x) \, dx$. Various other notations are also used sometimes. $\endgroup$ – littleO Nov 1 '17 at 13:01
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    $\begingroup$ As long as both $D$ and $f$ are clear from the context, it is perfectly acceptable and in fact is widely used throughout literature. $\endgroup$ – Sangchul Lee Nov 1 '17 at 13:25
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Normally one would use $dV$ as the measure indicator, that is

$$\int_D f(x) dV$$

I even think this is a better notation as the multiple integral notation suggests a specific order of integration which is not generally guaranteed to be the same (unless the requirements of Rubini's theorem is fulfilled).

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I'll be very sketchy. Under some assumptions (for example the integral if finite or the integrand function is nonnegative) the Fubini-Tonelli theorem ensures that $$\int_Y\left(\int_Xf(x,y)dx\right)dy = \int_{X\times Y} f(x,y) dx\otimes dy$$ where $dx\otimes dy$ is the product measure of $dx$ and $dy$. Usually in $\mathbb{R}^n$ with the Lebesgue measure $dx_1\cdot\cdot dx_n $ or $vol^n$ denotes the product measure of the $dx_i$ so when you write $\int_{J}\left(\int_I f(x,y) dx(x)\right) dy(y) $ (if you can use Fubini-Tonelli) you are writing $\int_{I\times J}f(w)vol^2(w)=\int_{I\times J}f(w)dxdy(w)$ .

On sets $D$ more general than $I\times J$ Fubini-Tonelli (if you can apply it) tells that $$\int_D f(x,y)dx\otimes dy(x,y)=\int_{\mathbb{R}}\left(\int_{D_y}f(x,y)dx(x)\right)dy(y)$$ where $D_y=\{x\in \mathbb{R} | \ (x,y)\in D\}$ Obviously you can write $\int_D f(x,y)dx\otimes dy(x,y)$ as $\int_D f(\chi)dx\otimes dy(\chi)$ or $\int_D f(\chi)vol^2(\chi)$.

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