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A finite increasing sequence of points that subdivides a closed interval.
A partition of an interval is a finite sequence of real numbers
The associated subintervals are for .
Partitions of the interval are used to build Riemann sums and the upper/lower sums that define the Riemann integral.
A Riemann sum of a bounded function with respect to a tagged partition , where , is the number
Riemann sums approximate the Riemann integral; the approximation is controlled by the mesh becoming small.
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 .