upper integral

Every upper Darboux sum is an over-estimate of the area, and finer partitions give smaller over-estimates. The upper integral is the best over-estimate of all: the smallest value the upper sums ever approach as you refine without limit. It is the tightest ceiling that all upper sums respect.

For a bounded f on [a, b], the upper integral is the infimum of upper Darboux sums over all partitions P: upper integral of f = inf_P U(f, P). This infimum exists and is finite because the set of upper sums is bounded below (every lower sum is a lower bound for it), so the completeness of the real numbers supplies the value.

The upper integral always exists for a bounded function, even when the ordinary integral does not. It can sit strictly above the lower integral; the two coincide exactly when f is Riemann integrable, and their common value is then the integral. So the upper integral is a one-sided approximant that becomes the real thing only when it meets its lower counterpart.