I'm beginning analytic number theory and I see this formula in Apostol's book : If $f$ has a continuous derivative $f'$ on the interval $[y,x]$, where $0 < y < x$, then $$ \sum_{y < n \le x} f(n) = \int_y^x f(t) \, dt + \int_y^x (t-[t]) f'(t) \, dt + f(x) ([x] - x) - f(y) ([y]-y). $$ The proof in Apostol's can be followed easily if one uses Riemann Integration. But since I meet with number theorists often I see more this kind of notation : $$ \sum_{y < n \le x} f(n) = \int_y^x f(t) d[t] = \text{something here I don't recall} - \int_y^x [t]f'(t) dt $$ because for some reason they can "integrate $d[t]$ and it gives $[t]$", which I don't understand, and I also don't really understand precisely what $d[t]$ stands for. I have done a measure theory course ; what I'm saying is that I don't understand all the details ; I understand that they "integrate by parts with the measure $d[t]$" which makes the proof quite simpler, but I don't understand the assumptions they make and how the details work out. I think that $d[t]$ could be a measure such that for $E \subseteq \mathbb R$ or $\mathbb C$, $d[t](E) = | \mathbb N \cap E |$, but I'm not sure.
Here's what I'm looking for : I don't want an intuitive point of view with plots or summations ; I want a formal proof from the viewpoint of a measure theorist, with details. Is there anyway this can be made clear? The reason why I want this is because I don't have much faith in the "integration by parts with $d[t]$" version of the proof, but number theorists seem to love it so much and they all sketch it ; I never managed to do it myself formally, even though I did a measure theory course.
Thanks for the help,
