Can someone show me why mathematicians use $d\mu$ instead of $dx$ for Lebesgue Integral over $u(x)$

Question

I am an engineer and I learned my Lebesgue integral from an engineering text which dumbed down a lot of stuff, most prominently all Lebesgue integrals were introduced as $\int_\Omega u(x) dx$ instead of $\int_\Omega u d\mu$.

I was basically told not to worry about it and just keep in mind if you are integrating over a point, then the measure of that point is zero hence the integral is zero. And I was assured that in most applications, Riemmanian Integration and Lebesgue Integration yields completely identical answers.

But now I am going through some stuff written by mathematicians and $d\mu$ is almost always used in place of $dx$ i.e. these notes. Is there any reason why I should care about this distinction?

Answer 1

It's a way of emphasizing that you're doing measure theory and using the Lebesgue integral, which is substantially more general than the Riemann integral. Among other things, it depends on a choice of measure (this is what $\mu$ refers to), and while choosing the Lebesgue measure reproduces the familiar answers you're used to from calculus, choosing other measures does other stuff. The $d \mu$ notation also continues to apply to multivariate integrals, whereas $dx$ really only applies to integrals over $\mathbb{R}$.

Answer 2

In some cases I think that the answer is easier than we believe. As somebody said, the abstract integral of a function $f$ with respect to a measure space $(X,\Sigma,\mu)$ is denoted by $\int_X u \, d\mu$. This is the most economic piece of notation. For particular purposes, we may felle the need to introduce a dummy integration variable, and we write $\int_X u(x)\, d\mu(x)$.

Since the Lebesgue measure is particularly useful in mathematical analysis and in many other fields, mathematicians decide to save letters and ink, and they write $dx$ instead of $d\mathcal{L}(x)$. In higher dimension, the notation $d\mathcal{L}^N(x)$ is seldom seen, and everybody writes $dx_1\cdots dx_N$.

Of course all this comes from another little abuse of notation: we should not write $$ \int_{\mathbb{R}^2} e^{-x^2-y^2}\, dx\, dy $$ but instead

$\int_{\mathbb{R}^2} f \, d\mathcal{L}^2$, where $f(x,y)=e^{-x^2-y^2}$.

We all understand why we prefer the abuse over the more correct version.