Question about relation between integral and summation in this case

Question

I have $N(t)=N_{\alpha_{n}}(t)=\#\lbrace n:\alpha_{n}\leq t\rbrace$.

Let $\{\alpha_{n}\}$ be a positive sequence, tending to infinity. Let $ \varphi (t) $ be a differentiable, positive, and non-increasing function on $[1;0]$, such that $ \varphi (t)\rightarrow 0 $.

Why do we have: for any $ z\geq 1 $,

$$\sum_{k:\, 1\leq \alpha_{k}\leq z} \varphi (\alpha_{k}) = \int_1^z \varphi (t) dNt\,?$$ Why do they use $dN(t)$?

Answer 1

Hint.

You should use the integration by parts for the Riemann-Stieljes integral:

$$\int_1^z \varphi(t) dN(t)=\varphi(z) N(z) - \varphi(1) N(1)- \int_1^z \varphi^\prime(t) N(t)dt$$ And using the fact that $N$ is piecewise constant.