Partition of an interval

A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with one subinterval indicated in red.

In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x = ( x_i )_i=1..k of real numbers such that

a = x₀ < x₁ < x₂ < ... < x_k = 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 [x_i, x_i+1] is referred to as a sub-interval 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.^[1]

Norm of a partition

The norm (or mesh) of the partition

x₀ < x₁ < x₂ < ... < x_n

is the length of the longest of these subintervals,^[2]^[3] that is

max{ (x_i − x_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.^[4]

Tagged partitions

A tagged partition^[5] is a partition of a given interval together with a finite sequence of numbers t₀, ..., t_n−1 subject to the conditions that for each i,

x_i ≤ t_i ≤ x_i+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 $x_{0},\ldots ,x_{n}$ together with $t_{0},\ldots ,t_{{n-1}}$ is a tagged partition of $[a,b]$ , and that $y_{0},\ldots ,y_{m}$ together with $s_{0},\ldots ,s_{{m-1}}$ is another tagged partition of $[a,b]$ . We say that $y_{0},\ldots ,y_{m}$ and $s_{0},\ldots ,s_{{m-1}}$ together is a refinement of a tagged partition $x_{0},\ldots ,x_{n}$ together with $t_{0},\ldots ,t_{{n-1}}$ if for each integer $i$ with $0\leq i\leq n$ , there is an integer $r(i)$ such that $x_{i}=y_{{r(i)}}$ and such that $t_{i}=s_{j}$ for some $j$ with $r(i)\leq j\leq r(i+1)-1$ . Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.