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?
Best Answer
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 $\mathbb{R}$.