Motivating notation that looks like the probability of differentials The Elements of Statistical Learning exemplifies a notation (which I have occasionally seen elsewhere) that I would like clarification on. From page 18:
$$\operatorname{EPE}(f) = \int [y - f(x)]^2 \color{red}{Pr(dx,dy)}$$
I would have understood $\color{blue}{dG_{X,Y}(x,y)}$ or $\color{magenta}{g_{X,Y}(x,y) dx\ dy}$ where $G$ and $g$ are the CDF and PDF respectively. When it comes to differentials I see them in some contexts as very tiny changes, and in others I think about them as unit basis vectors in tangent space. I keep interpreting $\color{red}{Pr(dx,dy)}$ as the probability of such tangent vectors, but I cannot seem to make any useful sense of that in this context.
Since the text also tells us that we are considering a statistical expectation, it must be equivalent to $\color{blue}{dG_{X,Y}(x,y)}$ and $\color{magenta}{g(x,y) dx\ dy}$ under an assumption of smoothness of the CDF. Without an assumption of smoothness we could have also written $$\int_0^{\infty}[y - f(x)]^2(1- G(x,y))dx\ dy - \int_{-\infty}^0 [y - f(x)]^2G(x,y) dx\ dy.$$
Is there an intuition or motivation behind this notation?

I would agree that it is equivalent with $dG_{X,Y}(x,y)$ but the notation $g(x,y)dx\,dy$ presumes that the PDF $g(x,y)$ exists, which is not always the case.
I think it is just notation for integration over the joint distribution.
