# Partition of an interval

In mathematics, a **partition** of an interval [*a*, *b*] on the real line is a finite sequence *x = ( x _{i} )* of real numbers such that

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

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## 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]} 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 *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.^{[citation needed]}

Suppose that together with is a tagged partition of , and that together with is another tagged partition of . We say that and together is a **refinement of a tagged partition** together with if for each integer with , there is an integer such that and such that for some with . Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

## See also

## 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. - ↑ 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.