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In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a partition of an interval on the real line is a finite sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of real number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s 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 , bis 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 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 apartial order
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
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
An integer is the number zero (), a positive natural number
In mathematics, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples are the natural number ...
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.
See also
* Regulated integral * Riemann integral * Riemann–Stieltjes integral * Partition of a setReferences
Further reading
* {{cite book , last=Gordon , first=Russell A. , title=The integrals of Lebesgue, Denjoy, Perron, and Henstock , series= Graduate Studies in Mathematics, 4 , publisher=American Mathematical Society , location=Providence, RI , year=1994 , isbn=0-8218-3805-9 Mathematical analysis