lower integral
Every lower Darboux sum is an under-estimate of the area, and finer partitions give larger under-estimates. The lower integral is the best under-estimate of all: the largest value the lower sums ever approach as you refine without limit. It is the highest floor that all lower sums respect.
For a bounded f on [a, b], the lower integral is the supremum of lower Darboux sums over all partitions P: lower integral of f = sup_P L(f, P). The supremum exists and is finite because the lower sums are bounded above (every upper sum is an upper bound for them), so completeness of the reals supplies the value.
The lower integral always exists for a bounded function and never exceeds the upper integral: lower integral <= upper integral. When the two are equal the function is Riemann integrable and the common number is the integral. The lower integral is the largest area a staircase strictly under the graph can capture, and integrability is exactly the assertion that nothing of substance is lost in the gap above it.
The inequality lower integral <= upper integral is not obvious from the definitions alone; it follows from the lemma that any lower sum is at most any upper sum, proved by comparing both against their common refinement.