In mathematics, a partition of an interval on the real line is a finite sequence of real numbers such that
.
In other terms, a partition of a compact interval is a strictly increasing sequence of numbers (belonging to the interval itself) starting from the initial point of and arriving at the final point of .
Every interval of the form is referred to as a subinterval of the partition x.
Another partition of the given interval [a, b] is defined as a refinement of the partition, if contains all the points of and possibly some other points as well; the partition is said to be “finer” than . Given two partitions, and, one can always form their common refinement, denoted, which consists of all the points of and, in increasing order.[1]
The norm (or mesh) of the partition
is the length of the longest of these subintervals[2] [3]
.
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]
A tagged partition[5] or Perron Partition is a partition of a given interval together with a finite sequence of numbers subject to the conditions that for each,
.
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 together with 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.