I have an integral over $t$ where I choose $t$ takes discrete values $t=0,1,2,...$ and would like to write the integral as a sum using the same function. I realize that this is not necessarily possible unless one is using a counting measure or something like that.
In the case of a one-dimensional Riemann sum over an interval from analysis, one can write
$\int^b_af(x) dx =\lim_{||\Delta x \rightarrow 0||} \sum_{i=1}^n f(x^*_i) \Delta x_i.$
Are there are other ways the integral can be changed to a sum? (I mean an actual equality and not an approximation with a remainder error term, although it could be something where the error term vanishes in the limit).