The unit step function $I$ is defined by
$ I(x)= \begin{cases}0,\quad x \le 0, \\ 1,\quad x>0. \end{cases} $
Let $f$ be continuous on $[a,b]$ and suppose $c_n\geq 0$ for $n=1, 2, 3,\ldots$ and $\sum_n c_n$ is convergent. Let $\alpha=\sum_{n=1}^{N} c_n I(x-s_n)$ where ${s_n}$ is a sequence of distinct points in $(a,b)$. Then $ \int_{a}^{b}fd\alpha=\sum_{i=1}^{N}c_n f(s_n). $
I can't understand why the last equation holds. Where does $f(s_n)$ comes from?