Question about integral and unit step function

Solution 1:

Remember the definition of the Riemann-Stieltjes integral. In the present example, the integrator $\alpha$ is constant between two consecutive $s_n$, hence two consecutive points from a partition of $[a,b]$ only contribute to the corresponding Riemann-Stieltjes sum if there is a $s_n$ between them, i. e. if $\alpha$ has a jump in between them. Their contribution is exactly the sum of all jumps happening. If the partition is fine enough, at most one jump happens between two neighboring points of the partition.

Solution 2:

$$ \begin{eqnarray} \int_{a}^{b}f(x)d\alpha(x)&=&\int_{a}^{b}f(x)d\left(\sum_{n=1}^{N} c_n I(x-s_n)\right)\\ &=&\sum_{n=1}^{N}c_n\int_{a}^{b}f(x)d\left( I(x-s_n)\right)\\ &=&\sum_{n=1}^{N}c_n\int_{a}^{b}f(x)\frac{d\left( I(x-s_n)\right)}{dx}dx\\ &=&\sum_{n=1}^{N}c_n\int_{a}^{b}f(x)\delta(x-s_n)dx\\ &=&\sum_{i=1}^{N}c_n f(s_n) \end{eqnarray} $$