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 âÂÂfinerâ than . Given two partitions, and , one can always form their common refinement, denoted , which consists of all the points of and , in increasing order.
The norm (or mesh) of the partition
is the length of the longest of these subintervals
Partitions are used in the theory of the Riemann integral, the RiemannâÂÂStieltjes 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.
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.