Integral of eigenkets (QM) I'm reading Dirac's book about QM. I reached the point (in my edition on page 37) where he tells it is possible, given the eigenkets of an observable, express any other ket in function of them (as he said before, for the definition of observable, its eigenkets form a complete set). So he writes this:

Let us examine mathematically the condition for a real dynamical variable $\xi$ to be an observable. Its eigenvalues may consist of a (finite or infinite) discrete set of numbers or, alternatively, they may consist of all numbers in a certain range, such as all numbers lying between $a$ and $b$. In the former case, the condition that any state is dependent on eigenstates of $\xi$ is that any ket can be expressed as a sum of eigenkets of $\xi$. In the latter case the condition needs modification, since one may have an integral instead of a sum, i.e. a ket $| P \rangle$ may be expressible as an integral of eigenkets of $\xi$,
$$ | P \rangle = \int | \xi' \rangle d\xi' $$

I would expand this infinite sum as follows (considering the general case where there are multiple eigenkets associated with a specific eigenvalue):
$$ | P \rangle = \int | \xi' \rangle d\xi' = \xi' |\xi'_1 \rangle + \xi' |\xi'_2 \rangle + \cdots + \xi' |\xi'_n \rangle  + \xi'' |\xi''_1 \rangle + \xi'' |\xi''_2 \rangle + \cdots + \xi'' |\xi''_m \rangle + \cdots +  \xi^{(j)} |\xi^{(j)}_1 \rangle + \xi^{(j)} |\xi^{(j)}_2 \rangle + \cdots + \xi^{(j)} |\xi^{(j)}_i \rangle$$
Using $n, m, j, i$ as arbitrary values. So I understand that if the eigenkets, due to hypothesis, form a base, it is possible to express any ket using a linear combination of them, but in a linear combination there must appear some coefficients, which inside this integral don't appear. And, also, how is it possible to integrate an eigenket with respect of its eigenvalue? They aren't strictly connected, are they?
More, he gives then this other expression:
$$ | P \rangle = \int | \xi'_c \rangle d\xi' + \sum_r | \xi^r_d \rangle $$
I interpret this based on the problems I had with the previous one?
 A: Recall in the book you are reading  PAMD has studiously avoided discussing the normalization of these kets until later, probably at the bottom of page 38, and then 39, when he introduces normalized $|A\rangle$. Read on.
There is absolutely nothing stopping you from incorporating the non vanishing numerical coefficients $\xi',\xi'', ...$ into the basis ket vectors they precede, so
$$ | P \rangle =    \xi' |\xi'_1 \rangle + \xi' |\xi'_2 \rangle + \cdots + \xi' |\xi'_n \rangle \\ + \xi'' |\xi''_1 \rangle + \xi'' |\xi''_2 \rangle + \cdots + \xi'' |\xi''_m \rangle  + \cdots +  \xi^{(j)} |\xi^{(j)}_1 \rangle + \xi^{(j)} |\xi^{(j)}_2 \rangle + \cdots + \xi^{(j)} |\xi^{(j)}_i \rangle\\ \mapsto 
   |\xi'_1 \rangle +  |\xi'_2 \rangle + \cdots + |\xi'_n \rangle   +   |\xi''_1 \rangle +   |\xi''_2 \rangle + \cdots +   |\xi''_m \rangle  + \cdots +   |\xi^{(j)}_1 \rangle +  |\xi^{(j)}_2 \rangle + \cdots +   |\xi^{(j)}_i \rangle .$$
The newly  rescaled basis vectors are again eigenvectors of $\xi$ labelled through them, with the very same numerical eigenvalues $\xi_1',...$ displayed inside the ket demarcator, |.
Just as you summed the denumerable basis vectors, you may now "sum" (integrate) the eigenvectors with continuous eigenvalues $\xi'$ — whose eigenvectors "know" about them: they are, in fact, functions of them.  PAMD helps you by contrasting them through the subscripts c and d, for "continuous and "discrete".
