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In mathematics, a **partition** of an interval [*a*, *b*] on the real line is a finite sequence *x*_{0}, *x*_{1}, *x*_{2}, …, *x _{n}* 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 **subinterval** of the partition *x*.

Another partition Q of the given interval [a, b] is defined as a **refinement of the partition** P, if Q 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, in increasing order.^{[1]}

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*}.

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]} or Perron Partition 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 *x*_{0}, …, *x _{n}* together with

**^**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.

- 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.