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I am working through the book "Elements of Statistical Learning" and I do not understand why $Pr(dx,dy)$ is in equation 2.10

Background:

enter image description here

This is what I understand so far:

I am calculating the expectation value of the loss function $L(Y,f(x))$, thus:

$$ \begin{eqnarray} E[L(Y,f(x))] &=& \int L(Y,f(x))\,\, Pr(X,Y)\,dxdy \label{} \\ &=&\int[y-f(x)]^2\,Pr(x,y)\,dxdy \end{eqnarray} $$

The question: How does $Pr(dx,dy)$ appear in the above equation, e.g. what steps are taken to arrive at the equality below: $$ \begin{eqnarray} \int[y-f(x)]^2\,Pr(x,y)\,dxdy = \int[y-f(x)]^2\,Pr(dx,dy) \end{eqnarray} $$

Related posts I've read:

  1. Expected Error Prediction Derivation

    This post says $Pr(x,y)\,dxdy = f_2(x,y)dxdy$

    Is this just standard notation, i.e. that the derivative of a joint distribution function $Pr(x,y)dxdy$ is written as $Pr(dx,dy)$ or $f_2(x,y)dxdy$. In which case is $Pr(x,y) = f_2(x,y)$? So: $$ \begin{eqnarray} Pr(x,y)\,dxdy = Pr(dx,dy) \end{eqnarray} $$

    If this is not just notation, is there a mathematical derivation for $$ \begin{eqnarray} \int[y-f(x)]^2\,Pr(x,y)\,dxdy = \int[y-f(x)]^2\,Pr(dx,dy) \end{eqnarray} $$

Edit

Changed notation $f(x,y)$ to $f_2(x,y)$ to clarify it is distinct from $f(x)$

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    $\begingroup$ First of all, you use the notation $f$ twice. Once for the statistical "model" $f(X)$ that should be close to $Y$, and once for the resulting joint density of the random variables $X$ and $Y$. That should be avoided. Secondly, assuming $X$ and $Y$ are continuous, it is indeed the case that $Pr(dx,dy)$ in the book really means $Pr(x\leqslant X<x+dx, y\leqslant Y<y+dx) = g_{XY}(x,y) dx dy$, where $g_{XY}(x,y)$ is the joint density of $X$ and $Y$. It is a bit of an unfortunate notation from a strictly calculus point-of-view, but a useful shorthand in probability. $\endgroup$
    – StijnDeVuyst
    Commented Jan 17, 2017 at 11:57
  • $\begingroup$ Yeah I know I used $f$ twice, the related post used $f(x,y)$ which really confused me. I replicated it in case it was intended. Thanks for the explanation regarding the notation, much appreciated! $\endgroup$
    – i_love_chocolate
    Commented Jan 17, 2017 at 20:28
  • $\begingroup$ This issue is discussed in several posts on prediction limits. In particular, see the extended discussions of specific prediction limits or intervals at stats.stackexchange.com/questions/4174, stats.stackexchange.com/questions/134380, and stats.stackexchange.com/questions/255570. $\endgroup$
    – whuber ♦
    Commented Jan 17, 2017 at 23:22

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