Rigorizing an interesting generalization of integration: $\lim_{\epsilon\to0}\sum_{n}g(n\epsilon,\epsilon)$

Question

Consider the following interesting formula:

$\lim_{\epsilon\to0}\sum_{n\in[0,1/\epsilon]}g(n\epsilon,\epsilon)$.

Consider the special case: $g(x,\epsilon)=\epsilon f(x)$, we have $\lim_{\epsilon\to0}\sum_{n\in[0,1/\epsilon]}g(n\epsilon,\epsilon)=\int_{0}^{1}f(x)dx$.

My question: can this class of generalization be made rigorous and well-defined? What are some books that I can read for the related problems?