What does Pr(dx, dy) mean? The book The Elements of Statistical Learning by Hastie and others (page 18) defines the expected value of prediction error as \begin{align} \operatorname{EPE}(f) &= \operatorname E(Y - f(X))^2\\\ & = \int [y - f(x)]^2 \Pr(dx, dy) \end{align} Why is it like above? Why not as below to be consistent with any expected value definition? $$ \operatorname{EPE}(f) = E(Y - f(x))^2 = \iint [y - f(x)]^2 \Pr(x,y) d(x) d(y)$$ What does $\Pr(dx, dy)$ even mean?

We are dealing with a probability density function. Since this is a bivariate density function our function outputs density per unit area (hence why an output $f(x,y)$ is not a valid probability measure on its own, and there is no restriction that $f(x,y) \leq 1$).

So we can argue for _very small_ $\Delta x$ and $\Delta y$, $\Pr(\Delta x, \Delta y) \approx f(x,y) \, \Delta x \, \Delta y$ and in the limit as $\Delta x$ and $\Delta y$ go to 0 we have $\Pr(dx, dy) = f(x,y)\,dx\,dy$.