Partition of an interval

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 = ( x_i ) of real numbers such that

a = x₀ < x₁ < x₂ < ... < 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.

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 < ... < 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 t₀, ..., 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 $x_0,\ldots,x_n$ together with $t_0,\ldots,t_{n-1}$ is a tagged partition of $[a, b]$, and that $y_0,\ldots,y_m$ together with $s_0,\ldots,s_{m-1}$ is another tagged partition of $[a,b]$. We say that $y_0,\ldots,y_m$ and $s_0,\ldots,s_{m-1}$ together is a refinement of a tagged partition $x_0,\ldots,x_n$ together with $t_0,\ldots,t_{n-1}$ if for each integer $i$ with $0 \le i \le n$, there is an integer $r(i)$ such that $x_i = y_{r(i)}$ and such that $t_i = s_j$ for some $j$ with $r(i) \le j \le r(i+1)-1$. Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

See also

• Regulated integral
• Riemann integral
• Riemann–Stieltjes integral
• Partition of a set

References

1. Brannan, D.A. (2006). A First Course in Mathematical Analysis. Cambridge University Press. p. 262. ISBN 9781139458955.
2. Hijab, Omar (2011). Introduction to Calculus and Classical Analysis. Springer. p. 60. ISBN 9781441994882.
3. Zorich, Vladimir A. (2004). Mathematical Analysis II. Springer. p. 108. ISBN 9783540406334.
4. Limaye, Balmohan (2006). A Course in Calculus and Real Analysis. Springer. p. 213. ISBN 9780387364254.
5. 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.