You are right. I didn't need to look at the Poisson formula. I guess I was fixated on that because of the Hilbert transform I was also looking at. In the end I started going back at some definitions after the comments and realized, my confusion came from the fact that the papers I was reading were not careful defining the Hilbert transform and omitted the fact, that the integrals were principal valued.

I know this is a standard fact. My point is that I have found countless times (such as sciencedirect.com/topics/mathematics/hilbert-transform or sciencedirect.com/topics/psychology/hilbert-transform) that the Fourier transform of $1/x$ in the usual sense is being used as the sign function. My question is exactly why that is and I would have settled for a Cauchy Principal Value kind of integral, but many authors seem to imply otherwise.

Thank you! I'm posting the link here for those who want to take a look too. comp.tmu.ac.jp/kzmurota/paper/HIMSummerSchool15Murota.pdf

Thank you very much, I guess this is the integral I was looking for

@losifPInelis do you know of any good references to learn more about the desintegration of a measure?

@CarloBeenakker, how do you define the delta function measure? This seems like what I'm looking for, but I'm unsure of how to justify my decision of measure.

I'm not sure I understand. What would be, for example, the measure from the below example and my first method and how does it differ from the Lebesgue measure? What is then the measure from the second method? And thank you very much for the comment, your answers always help a lot.

I'm confused now, because this also differs from the answer form @CarloBeenakker. I indeed want to integrate over a measure induced by the Lebesgue measure, but do you know why all the answers are different? The presence or absence of the factor $\frac{1}{\sqrt{k}}$ is importante for me.

I'm going to check my iterations for this mistake! Thanks! It is good to know, that method 1 works. Do you have any idea what went wrong with method 2?

It's because $(\vec{y},Q\vec{y})=\sum_{i=1}^{k}y_i^2-\frac{1}{k}(\sum_{i=1}^{k}y_i)^2$ and we are subject to the bound $\sum_{i=1}^{k}y_i=x$