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| In [[mathematics]], a '''partition''', ''P'' of an [[interval (mathematics)|interval]] [''a'', ''b''] on the [[real number|real]] line is a finite [[sequence]] of the form
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| :''a'' = ''x''<sub>0</sub> < ''x''<sub>1</sub> < ''x''<sub>2</sub> < ... < ''x''<sub>''n''</sub> = ''b''.
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| Another partition of the given interval, ''Q'', is defined as a '''refinement of the partition''', ''P'', when it contains all the points of ''P'' and possibly some other points as well; the partition ''Q'' is said to be “finer” than ''P''. Given two partitions, ''P'' and ''Q'', one can always form their '''common refinement''', denoted ''P'' ∨ ''Q'', which consists of all the points of ''P'' and ''Q'', re-numbered in order.<ref>{{cite book|author=Brannan, D.A.|title=A First Course in Mathematical Analysis|publisher=Cambridge University Press|year=2006|isbn=9781139458955|page=262|url=http://books.google.com/books?id=N8bL9lQUGJgC&pg=PA262}}</ref>
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| The '''norm''' (or '''mesh''') of the partition
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| :''x''<sub>0</sub> < ''x''<sub>1</sub> < ''x''<sub>2</sub> < ... < ''x''<sub>''n''</sub>
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| is the length of the longest of these subintervals,<ref>{{Cite book|author=Hijab, Omar|title=Introduction to Calculus and Classical Analysis|publisher=Springer|year=2011|isbn=9781441994882|page=60|url=http://books.google.com/books?id=_gb9fMqur9kC&pg=PA60}}</ref><ref>{{Cite book|author=Zorich, Vladimir A.|title=Mathematical Analysis II|publisher=Springer|year=2004|isbn=9783540406334|page=108|url=http://books.google.com/books?id=XF8W9W-eyrgC&pg=PA108}}</ref> that is | |
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| :max{ |''x''<sub>''i''</sub> − ''x''<sub>''i''−1</sub>| : ''i'' = 1, ..., ''n'' }.
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| 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|author=Limaye, Balmohan|title=A Course in Calculus and Real Analysis|publisher=Springer|year=2006|isbn=9780387364254|page=213|url=http://books.google.com/books?id=Ou53zXSBdocC&pg=PA213}}</ref>
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| A '''tagged partition'''<ref>{{cite book|authors=Dudley, Richard M. & Norvaiša, Rimas|title=Concrete Functional Calculus|publisher=Springer|year=2010|isbn=9781441969507|page=2|url=http://books.google.com/books?id=fuuB59EiIagC&pg=PA2}}</ref> is a partition of a given interval together with a finite sequence of numbers ''t''<sub>0</sub>, ..., ''t''<sub>''n''−1</sub> subject to the conditions that for each ''i'',
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| :''x''<sub>i</sub> ≤ t<sub>i</sub> ≤ x<sub>i+1</sub>.
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| 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.
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| Suppose that <math>\scriptstyle x_0,\ldots,x_n</math> together with <math>\scriptstyle t_0,\ldots,t_{n-1}</math> is a tagged partition of <math>[a, b]</math>, and that <math>\scriptstyle y_0,\ldots,y_m</math> together with <math>\scriptstyle s_0,\ldots,s_{m-1}</math> is another tagged partition of <math>[a,b]</math>. We say that <math>\scriptstyle y_0,\ldots,y_m</math> and <math>\scriptstyle s_0,\ldots,s_{m-1}</math> together is a '''refinement of a tagged partition''' <math>\scriptstyle x_0,\ldots,x_n</math> together with <math>\scriptstyle t_0,\ldots,t_{n-1}</math> if for each integer <math>i</math> with <math>\scriptstyle 0 \le i \le n</math>, there is an integer <math>r(i)</math> such that <math>\scriptstyle x_i = y_{r(i)}</math> and such that <math>t_i = s_j</math> for some <math>j</math> with <math>\scriptstyle r(i) \le j \le r(i+1)-1</math>. Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.
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| == See also ==
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| * [[Regulated integral]]
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| * [[Riemann integral]]
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| * [[Riemann–Stieltjes integral]]
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| * [[Partition of a set]]
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| == References ==
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| {{Reflist}}
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| ==Further reading==
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| * {{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 }}
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| [[Category:Mathematical analysis]]
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