refinement
Refining a partition means cutting it further — keeping all the old division points and inserting new ones. Every strip either stays the same or gets split into thinner pieces. Intuitively a finer staircase tracks the curve more closely, so refinement is the engine that drives the Darboux sums toward each other.
Formally, a partition Q is a refinement of a partition P if P is a subset of Q, that is, every point of P is also a point of Q. Given any two partitions P_1 and P_2, their union P_1 union P_2 is a common refinement of both, which is the standard device for comparing sums built from unrelated partitions.
The key monotonicity fact is that refinement never makes the bracket worse: passing to a refinement can only raise (or keep) the lower sum and only lower (or keep) the upper sum, so L(f, P) <= L(f, Q) <= U(f, Q) <= U(f, P) whenever Q refines P. This squeezing is precisely why the lower and upper integrals are well defined and why every lower sum is dominated by every upper sum — pass to the common refinement and chain the inequalities.
Take P = {0, 1} and Q = {0, 1/2, 1} on [0, 1] with f(x) = x. Then L(f, P) = 0 and L(f, Q) = (0)(1/2) + (1/2)(1/2) = 1/4, so the lower sum rose; U(f, P) = 1 fell to U(f, Q) = (1/2)(1/2) + (1)(1/2) = 3/4.
Adding one midpoint pulls the lower sum up and the upper sum down, narrowing the bracket from [0, 1] to [1/4, 3/4].
Refinement is about adding points, not about mesh: a refinement always has mesh no larger than the original, but two partitions can refine each other's mesh without either refining the other as a point set. The Darboux theory is built on the point-set order, not the mesh order.