Partition of an interval

In mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the form

a = x₀ < x₁ < x₂ < ... < x_n = b.

Such partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral and the regulated integral. 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.

The norm (or mesh) of the partition

x₀ < x₁ < x₂ < ... < x_n

is the length of the longest of these subintervals, that is

max{ |x_i − x_i−1| : i = 1, ..., n }.

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.

A tagged 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.

Suppose that $\scriptstyle x_0,\ldots,x_n$ together with $\scriptstyle t_0,\ldots,t_{n-1}$ is a tagged partition of [a,b], and that $\scriptstyle y_0,\ldots,y_m$ together with $\scriptstyle s_0,\ldots,s_{m-1}$ is another tagged partition of [a,b]. We say that $\scriptstyle y_0,\ldots,y_m$ and $\scriptstyle s_0,\ldots,s_{m-1}$ together is a refinement of a tagged partition $\scriptstyle x_0,\ldots,x_n$ together with $\scriptstyle t_0,\ldots,t_{n-1}$ if for each integer i with $\scriptstyle 0 \le i \le n$, there is an integer r(i) such that $\scriptstyle x_i = y_{r(i)}$ and such that t_i = s_j for some j with $\scriptstyle r(i) \le j \le r(i+1)-1$. 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

References

• 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.