In mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the form
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
is the length of the longest of these subintervals, that is
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,
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 together with is a tagged partition of , and that together with is another tagged partition of . We say that and together is a refinement of a tagged partition together with if for each integer with , there is an integer such that and such that for some with . Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.