thumb|300px|A partition of an interval being used in a [[Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red.]]
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.
Refinement of a partition
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.
Norm of a partition
The norm (or mesh) of the partition
:
is the length of the longest of these subintervals
: }.
Applications
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.
Tagged partitions
A tagged partition 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.
See also
- Regulated integral
- Riemann integral
- Riemann–Stieltjes integral
- Henstock–Kurzweil integral