Partition of an interval

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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 = ( xi )i=1..k of real numbers such that

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 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 < ... < xn

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

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