# Partition of an interval

In mathematics, a **partition** of an interval [*a*, *b*] on the real line is a finite sequence *x*_{0}, *x*_{1}, *x*_{2}, ..., *x _{k}* of real numbers such that

*a*=*x*_{0}<*x*_{1}<*x*_{2}< ... <*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 **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.[1]

## Norm of a partition

The **norm** (or **mesh**) of the partition

*x*_{0}<*x*_{1}<*x*_{2}< ... <*x*_{n}

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

- 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 *t*_{0}, ..., *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}, ..., *x _{n}* together with

*t*

_{0}, ...,

*t*

_{n − 1}is a tagged partition of [

*a*,

*b*], and that

*y*

_{0}, ...,

*y*together with

_{m}*s*

_{0}, ...,

*s*

_{m − 1}is another tagged partition of [

*a*,

*b*]. We say that

*y*

_{0}, ...,

*y*and

_{m}*s*

_{0}, ...,

*s*

_{m − 1}together is a

**refinement of a tagged partition**

*x*

_{0}, ...,

*x*together with

_{n}*t*

_{0}, ...,

*t*

_{n − 1}if for each integer i with 0 ≤

*i*≤

*n*, there is an integer

*r*(

*i*) such that

*x*=

_{i}*y*

_{r(i)}and such that

*t*=

_{i}*s*for some j with

_{j}*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.

## References

- Brannan, D. A. (2006).
*A First Course in Mathematical Analysis*. Cambridge University Press. p. 262. ISBN 9781139458955. - Hijab, Omar (2011).
*Introduction to Calculus and Classical Analysis*. Springer. p. 60. ISBN 9781441994882. - Zorich, Vladimir A. (2004).
*Mathematical Analysis II*. Springer. p. 108. ISBN 9783540406334. - Ghorpade, Sudhir; Limaye, Balmohan (2006).
*A Course in Calculus and Real Analysis*. Springer. p. 213. ISBN 9780387364254. - Dudley, Richard M.; Norvaiša, Rimas (2010).
*Concrete Functional Calculus*. Springer. p. 2. ISBN 9781441969507.

## Further reading

- Gordon, Russell A. (1994).
*The integrals of Lebesgue, Denjoy, Perron, and Henstock*. Graduate Studies in Mathematics, 4. Providence, RI: American Mathematical Society. ISBN 0-8218-3805-9.