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Refining a partition decreases upper sums and increases lower sums.
Refinement lemma: Let and let be bounded. If is a refinement of a partition , then
where and denote the upper sum and lower sum of with respect to .
This monotonicity under refinement underlies the definition of the Riemann integral as the common value of the infimum of upper sums and the supremum of lower sums.
A refinement of a partition of is a partition of such that every point of is also a point of . In other words, in the sense of subset.
Refinements are used to compare upper sums and lower sums: making a partition finer is the basic way to force these sums closer together in Riemann integrability.
A partition of an interval is a finite sequence of real numbers
The associated subintervals are for .
A lower sum of a bounded function with respect to a partition is the number
where is the infimum of on the th subinterval.
Lower sums are paired with upper sums to define Riemann integrability via the gap .
A Riemann integral of a Riemann integrable function is the number
defined by
where and are the lower and upper sums over all partitions of .