An intricate integral involving indicator functions

Question

Consider the following integral: $$\int_a^bf(x)\text{ }\mathrm{d}\phi(x)$$ where $\phi(x)$ is defined as: $$\phi(x)=A\sum_{i=1}^n\mathbb{1}(x_i<x)$$

Where the $x_i$'s are a sequence of known real numbers. How would you solve this?

Answer 1

$\phi$ is a superposition of Heavyside functions. Their derivative are Dirac's delta functions. When integrated, these delta functions translate to evaluation of the integrand function, so the solution is related to the values of $f$ at your points $x_i$.

As a side comment, you did not define the sequence $x_i$, I assume they are a known list of real numbers.