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 = x₀ < x₁ < x₂ < … < x_n = b.

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.

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 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.^[5]

See also

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