In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x0, x1, x2, …, xn of real numbers such that
a = x0 < x1 < x2 < … < xn = b. In other terms, a partition of a compact interval I is a strictly increasing sequence of numbers (belonging to the interval I itself) starting from the initial point of I and arriving at the final point of I. Every interval of the form [xi, xi + 1] is referred to as a subinterval of the partition x.
Refinement of a partition Another partition Q of the given interval [a, b] is defined as a refinement of the partition P, if Q contains all the points of P and possibly some other points as well; the partition Q is said to be “finer” than P. Given two partitions, P and Q, one can always form their common refinement, denoted P ∨ Q, which consists of all the points of P and Q, in increasing order.
Norm of a partition The norm (or mesh) of the partition
x0 < x1 < x2 < … < xn is the length of the longest of these subintervals
max{|xi − xi−1| : i = 1, … , n }.
Applications Partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral and the regulated integral. Specifically, as finer partitions of a given interval are considered, their mesh approaches zero and the Riemann sum based on a given partition approaches the Riemann integral.
Tagged partitions A tagged partition or Perron Partition is a partition of a given interval together with a finite sequence of numbers t0, …, tn − 1 subject to the conditions that for each i,
xi ≤ ti ≤ xi + 1. In other words, a tagged partition is a partition together with a distinguished point of every subinterval: its mesh is defined in the same way as for an ordinary partition.
See also Regulated integral Riemann integral Riemann–Stieltjes integral Henstock–Kurzweil integral
References
Further reading Gordon, Russell A. (1994). The integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics, 4. Providence, RI: American Mathematical Society. ISBN 0-8218-3805-9.
