The theorem says:
Suppose $c_n \geq 0$ for $1,2,3 ...$. $\sum c_n$ converges, $\{s_n\}$ is a sequence of a distinct points in $(a,b)$, and $\alpha (x) = \sum^{\infty}_{n=1} c_n I(x-s_n)$. Let $f$ be continuous on $[a,b]$. Then $$\int ^b _a f d\alpha = \sum^{\infty}_{n=1} c_n f(s_n).$$
So I was wondering does $\{s_n\}$ have to be monotone? If not, the coefficient of each $f(s_n)$ in infinite series would not be a $c_n$. Instead, it would be the sum of several $c_n$'s which is the difference between $\alpha(s_i)$ and $\alpha(s_{i-1})$. Am I understanding it wrong?