TITLE: Never seen this notation before: $\int (y-f(x))^2 Pr(dx,dy) $
QUESTION [8 upvotes]: I have never seen an integral like this:
$$\int (y-f(x))^2 Pr(dx,dy) $$
What is that? More precisely what is $Pr(dx,dy)$? And how is that integral defined? I found it in Elements of Statistical Learning, By T. Hastie, et. al.

REPLY [4 votes]: $Pr(dx,dy)$ is the joint distribution. For example, if the density of the joint distribution is $g(x,y)$ , then
$$
\int (y-f(x))^2 Pr(dx,dy)=\int\int (y-f(x))^2 g(x,y)\,dx\,dy,
$$
over your region of integration.