Partition of an interval

From Wikipedia, the free encyclopedia

In mathematics, a partition 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.

The norm (or mesh) of the partition

x₀ < x₁ < x₂ < ... < x_n

is the length of the longest of these subintervals; it is

max{ |x_i − x_i−1| : i = 1, ..., n }.

As the mesh approaches zero, a Riemann sum based on the partition approaches the Riemann integral.

A tagged partition is a partition of an 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, it is a partition together with a distinguished point of every subinterval. The mesh of a tagged partition is defined the same as for an ordinary partition. We can 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 together with are a tagged partition of [a,b], and that together with are another tagged partition of [a,b]. We say that and together are a refinement of together with if for each integer i with , there is an integer r(i) such that x_i = y_r(i) and such that t_i = s_j for some j with . Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

[edit] 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.