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 . 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{{{abs|x{{sub|i}} − x{{sub|i−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. It is possible to define a partial order on the set of all tagged partitions by saying that one tagged partition is bigger than another if the bigger one is a refinement of the smaller one. Suppose that x0, …, xn together with t0, …, tn − 1 is a tagged partition of [a, b] , and that y0, …, ym together with s0, …, sm − 1 is another tagged partition of [a, b] . We say that y0, …, ym together with s0, …, sm − 1 is a refinement of a tagged partition x0, …, xn together with t0, …, tn − 1 if for each integer i with 0 ≤ i ≤ n , there is an integer r(i) such that and such that for some j with r(i) ≤ j ≤ r(i + 1) − 1 . Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.