Lower Darboux integral ($\underline{{\int }_a^b}f={\sup }_{P}L(f,P)$)

The lower Darboux integral of a bounded real function on a closed interval is the supremum of its lower Darboux sums over all partitions of an interval. Refinement increases a lower Darboux sum. A common refinement of two arbitrary partitions of an interval proves that every lower Darboux sum is at most every upper Darboux sum, hence the lower Darboux integral never exceeds the upper Darboux integral.