 | ||
In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x0, x1, x2, ..., xk of real numbers such that
Contents
a = x0 < x1 < x2 < ... < xk = 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 of the given interval, Q, is defined as a refinement of the partition, P, when it 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, re-numbered in order.
Norm of a partition
The norm (or mesh) of the partition
x0 < x1 < x2 < ... < xnis the length of the longest of these subintervals, that is
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 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 and s0, ..., sm − 1 together 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 xi = yr(i) and such that ti = sj 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.