Elements of Statistical Learning Integral Notation In equation 2.9 and 2.10 on page 18 of ESL we have
$$E(Y - f(X))^2 = \int [y - f(x)]^2 Pr(dx, dy)$$
However this notation confuses me. I'm rather expecting
$$E(Y - f(X))^2 = \int [y - f(x)]^2 Pr(x, y)dxdy$$
Is this just a matter of notation, or am I missing some more rigurous real analysis / probability theory explanation?
 A: Using $\int ...\,\text dx\text dy$ generally implies that we're integrating with respect to the Lebesgue measure, so in an integral like
$$
\int [y - f(x)]^2 P(x,y)\,\text dx\text dy
$$
this looks like we have a Lebesgue PDF $P(x,y)$ (while recognizing that this is an abuse of notation, albeit a common one) and are integrating $(y - f(x))^2 P(x,y)$ over $\mathbb R^2$ using the Lebesgue measure.
The notation
$$
\int [y - f(x)]^2 \,P(\text dx, \text dy)
$$
makes it explicit that we are integrating $(x, y) \mapsto [y - f(x)]^2$ over $\mathbb R^2$ using the measure $P$, so this is a Lebesgue integral, and there's no suggestion of densities or dominating measures. That makes this notation more acceptable. I think the intuition for this notation is that $\text dx$ and $\text dy$ are like small lengths so we can imagine plugging them into $P$ and measuring them, although it doesn't really mean that we're doing that and ultimately this is just a notation for a Lebesgue integral with the measure $P$.
An alternative notation that I think I see more often is
$$
\int [y-f(x)]^2 \,\text dP(x, y)
$$
where here even though $P$ is a measure and doesn't act directly on $x$ and $y$ (i.e. its domain is subsets of $\mathbb R$, not elements of $\mathbb R$), the $x$ and $y$ in the $\text dP$ are just to show that we're integrating over both of them.
