Partition of an interval

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 = 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.

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

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 }.

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 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.

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

• 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.