I have a (vector) space of step functions from the unit interval to the reals. Each function is discontinuous at a countable, possibly infinite, number of points. Since the discontinuities have measure zero, I can integrate the functions. I want a strict weak order in which one is less than another if its integral is less. However, some of the integrals diverge. What is the most natural extension of this ordering to these cases (or is there a reason why I should not expect any one to be more natural than another)?

I'm not really sure what field this is in. I've read about distributions, vector spaces, and measure theory, but I can't find anything that really applies. If you know of any related idea, I can try to make the question more precise.