Question
Let $\{s_n\},\{\sigma_n\}\subset\mathbb{R}$ be monotone increasing and are able to be summed. Also let $\{u_n\},\{v_n\}\subset\mathbb{R}$ and $A < u_i < v_i < u_{i+1} < B$ for all $i$. Let $H$ denote the Heaviside function and suppose that $$f(x) = \sum_{n=0}^\infty s_n H(x-u_n)$$ and $$g(x) = \sum_{n=0}^\infty \sigma_n H(x-v_n).$$ Express $\int_A^Bfdg$ as a sum using $\sigma_n$ and $\sum_{i=0}^ns_i$.
Attempt
I know we can break down the integral and interchange the sum and integral to get $$\sum_{n=0}^\infty \int_A^B s_n H(x-u_n)dg.$$ I'm trying to understand how to eliminate one of the Heaviside functions although it's not immediately clear if that's possible and how to achieve $\sum_{i=0}^ns_i$ in the final form.