This is equation 2.10 from the book Elements of Statistical Learning:
$$ E[(Y-f(X))^2] = \int (y-f(X))^2Pr(dx,dy) $$
$X$ is a real valued random input vector and $Y$ is a real valued random output. $X$ and $Y$ have joint distribution $Pr(X,Y)$. $f(X)$ is an approximation of $Y$ for a given $X$.
I am confused by the $dx$ and $dy$ within $Pr$ in the integral. Does $Pr(dx,dy)$ mean the probability of $X$ and $Y$ lying in the $dx$ and $dy$ ranges, respectively?
Is the integral equivalent to $$ E[(Y-f(X))^2] = \int (y-f(X))^2 p(x,y) dx,dy $$ where $p(x,y)$ is the joint pdf of $X$ and $Y$?