Editing Partition of an interval

{{Short description|Increasing sequence of numbers that span an interval}}
{{about|grouping elements of an interval using a sequence|grouping elements of a set using a set of sets|Partition of a set}}

[[File:Integral Riemann sum.png|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 (mathematics)|interval]] {{math|[''a'', ''b'']}} on the [[real line]] is a finite [[sequence]] {{math|''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x<sub>n</sub>''}} of [[real number]]s such that

:{{math|''a'' {{=}} ''x''<sub>0</sub> < ''x''<sub>1</sub> < ''x''<sub>2</sub> < … < ''x''<sub>''n''</sub> {{=}} ''b''}}.

In other terms, a partition of a [[compact space|compact]] interval {{mvar|I}} is a strictly increasing sequence of numbers (belonging to the interval {{mvar|I}} itself) starting from the initial point of {{mvar|I}} and arriving at the final point of {{mvar|I}}.

Every interval of the form {{math|[''x''<sub>''i''</sub>, ''x''<sub>''i'' + 1</sub>]}} is referred to as a '''subinterval''' of the partition ''x''.

==Refinement of a partition==
Another partition {{mvar|Q}} of the given interval [a, b] is defined as a '''refinement of the partition''' {{mvar|P}}, if {{mvar|Q}} contains all the points of {{mvar|P}} and possibly some other points as well; the partition {{mvar|Q}} is said to be “finer” than {{mvar|P}}. Given two partitions, {{mvar|P}} and {{mvar|Q}}, one can always form their '''common refinement''', denoted {{math|''P''&nbsp;∨&nbsp;''Q''}}, which consists of all the points of {{mvar|P}} and {{mvar|Q}}, in increasing order.<ref>{{cite book|last=Brannan|first=D. A.|title=A First Course in Mathematical Analysis|publisher=Cambridge University Press|year=2006|isbn=9781139458955|page=262|url=https://books.google.com/books?id=N8bL9lQUGJgC&pg=PA262}}</ref>

==Norm of a partition==
The '''norm''' (or '''mesh''') of the partition
: {{math|''x''<sub>0</sub> < ''x''<sub>1</sub> < ''x''<sub>2</sub> < … < ''x''<sub>''n''</sub>}}

is the length of the longest of these subintervals<ref>{{Cite book|last=Hijab|first=Omar|title=Introduction to Calculus and Classical Analysis|publisher=Springer|year=2011|isbn=9781441994882|page=60|url=https://books.google.com/books?id=_gb9fMqur9kC&pg=PA60}}</ref><ref>{{Cite book|author=Zorich, Vladimir A.|author-link=Vladimir Zorich|title=Mathematical Analysis II|publisher=Springer|year=2004|isbn=9783540406334|page=108|url=https://books.google.com/books?id=XF8W9W-eyrgC&pg=PA108}}</ref>
: {{math|max{{{abs|''x''<sub>''i''</sub> − ''x''<sub>''i''−1</sub>}} : ''i'' {{=}} 1, … , ''n'' }}}.

==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]].<ref>{{cite book|last1=Ghorpade|first1=Sudhir|last2=Limaye|first2=Balmohan|title=A Course in Calculus and Real Analysis|publisher=Springer|year=2006|isbn=9780387364254|page=213|url=https://books.google.com/books?id=Ou53zXSBdocC&pg=PA213}}</ref>

==Tagged partitions==
A '''tagged partition''' or Perron Partition is a partition of a given interval together with a finite sequence of numbers {{math|''t''<sub>0</sub>, …, ''t''<sub>''n'' − 1</sub>}} subject to the conditions that for each {{mvar|i}},
: {{math|''x<sub>i</sub>'' ≤ ''t<sub>i</sub>'' ≤ ''x''<sub>''i'' + 1</sub>}}.
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.<ref>{{cite book|last1=Dudley|first1=Richard M.|last2=Norvaiša|first2=Rimas|title=Concrete Functional Calculus|publisher=Springer|year=2010|isbn=9781441969507|page=2|url=https://books.google.com/books?id=fuuB59EiIagC&pg=PA2}}</ref>

==See also==
* [[Regulated integral]]
* [[Riemann integral]]
* [[Riemann–Stieltjes integral]]
* [[Henstock–Kurzweil integral]]

==References==
{{Reflist}}

==Further reading==
* {{cite book | last=Gordon | first=Russell A. | title=The integrals of Lebesgue, Denjoy, Perron, and [[Ralph Henstock|Henstock]] | series=[[Graduate Studies in Mathematics]], 4 | publisher=American Mathematical Society | location=Providence, RI | year=1994 | isbn=0-8218-3805-9 }}

[[Category:Mathematical analysis]]