Partition of an interval

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

a = x0 < x1 < x2 < ... < xn = b.

Such partitions are used in the theory of the Riemann integral, the Riemann-Stieltjes integral and the regulated integral. A refinement of a partition, P, is another partition, Q, of the given interval that 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 < ... < xn

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

max{ |xi − xi−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, ..., tn−1 subject to the conditions that for each i,

xi ≤ ti ≤ xi+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 ti = sj for some j with \scriptstyle r(i) \le j \le r(i+1). Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

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[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.