The thing is, you really don't neeed to consider non-Riemann integrable functions if you're actually trying to integrate or measure things. At some point, trying to find non-Borel measurable, but Lebesgue measurable sets serves no practical purpose other than being written down in an analysis textbook. More power to pure mathematicians and their students who enjoy playing these games, but the more applied folks can get around those "problems" by just making some niceness assumptions about your model (like f is Riemann integrable!) and moving on to the more pressing issues in their fields.