Partition of an interval

This article is about grouping elements of an interval using a sequence. For grouping elements of a set using a set of sets, see Partition of a set.

In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x₀, x₁, x₂, …, x_n of real numbers such that

A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red.

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 subinterval of the partition x.

Refinement of a partition

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]

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]

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] or Perron Partition 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.^{[citation needed]}

Suppose that x₀, …, x_n together with t₀, …, t_{n − 1} is a tagged partition of [a, b], and that y₀, …, y_m together with s₀, …, s_{m − 1} is another tagged partition of [a, b]. We say that y₀, …, y_m together with s₀, …, s_{m − 1} is a refinement of a tagged partition x₀, …, x_n together with t₀, …, 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_j for some j with 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.

See also

• Regulated integral
• Riemann integral
• Riemann–Stieltjes integral
• Henstock–Kurzweil integral

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. Ghorpade, Sudhir; 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.