# Partition of an interval

In mathematics, a partition, P, of an interval [a, b] on the real line is a finite sequence of the form

a = x0 < x1 < x2 < ... < xn = b.

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]

The norm (or mesh) of the partition

x0 < x1 < x2 < ... < xn

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

max{ |xixi−1| : i = 1, ..., n }.

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]

A tagged partition[5] 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.[citation needed]

Suppose that $\scriptstyle x_0,\ldots,x_n$ together with $\scriptstyle t_0,\ldots,t_{n-1}$ is a tagged partition of $[a, b]$, and that $\scriptstyle y_0,\ldots,y_m$ together with $\scriptstyle s_0,\ldots,s_{m-1}$ is another tagged partition of $[a,b]$. We say that $\scriptstyle y_0,\ldots,y_m$ and $\scriptstyle s_0,\ldots,s_{m-1}$ together is a refinement of a tagged partition $\scriptstyle x_0,\ldots,x_n$ together with $\scriptstyle t_0,\ldots,t_{n-1}$ if for each integer $i$ with $\scriptstyle 0 \le i \le n$, there is an integer $r(i)$ such that $\scriptstyle x_i = y_{r(i)}$ and such that $t_i = s_j$ for some $j$ with $\scriptstyle r(i) \le j \le r(i+1)-1$. Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.