I found in Conway's functional analysis there is a definition of spectral integral that looks more natural. I have a feeling that this is something similar to the definition of Riemann or Lebesgue integral. I think this might be a paralleled version of Riemann integral we learnt in calculus: "for all $\epsilon>0$, there exists $\delta>0$ such that for every partition of $[a,b]$ with norm $\|P\|<\delta$ and $\xi_i\in[x_{i-1},x_i]$ be the sample points, then $\left|\sum_{i=1}^n f(\xi_i)\Delta x_i-A\right|<\epsilon$". However, I can hardly seen the link between "norm condition" I wrote above and the $\sup\{\cdots\}$ condition in the figure below. Can anyone explain how to intutive understand the spectral integral?
