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:

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 $f(x)$

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:

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 $f(x)$

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:

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 $Pr(x,y)\,dxdy = f(x,y)dxdy$

   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 $f(x,y)dxdy$. In which case is $Pr(x,y) = f(x,y)$? So: $$ \begin{eqnarray} Pr(x,y)\,dxdy = Pr(dx,dy) \end{eqnarray} $$

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

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:

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 $f(x)$

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:  

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 $Pr(x,y)\,dxdy = f(x,y)dxdy$

   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 $f(x,y)dxdy$. In which case is $Pr(x,y) = f(x,y)$? So: $$ \begin{eqnarray} Pr(x,y)\,dxdy = Pr(dx,dy) \end{eqnarray} $$

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

Thanks in advance!

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: 

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 $Pr(x,y)\,dxdy = f(x,y)dxdy$

   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 $f(x,y)dxdy$. In which case is $Pr(x,y) = f(x,y)$? So: $$ \begin{eqnarray} Pr(x,y)\,dxdy = Pr(dx,dy) \end{eqnarray} $$

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