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. 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 {{math|[a, b]}} on the real line is a finite sequence {{math|x₀, x₁, x₂, …, x_n}} of real numbers such that

:{{math|a {{=}} x₀ < x₁ < x₂ < … < x_n {{=}} b}}.

In other terms, a partition of a 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_i, x_{i + 1}]}} 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 ∨ Q}}, which consists of all the points of {{mvar|P}} and {{mvar|Q}}, in increasing order.{{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}}

Norm of a partition

The norm (or mesh) of the partition

: {{math|x₀ < x₁ < x₂ < … < x_n}}

is the length of the longest of these subintervals{{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}}{{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}}

: {{math|max{{{abs|x_i − x_i−1}} : 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.{{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}}

Tagged partitions

A tagged partition or Perron Partition is a partition of a given interval together with a finite sequence of numbers {{math|t₀, …, t_{n − 1}}} subject to the conditions that for each {{mvar|i}},

: {{math|x_i ≤ t_i ≤ x_{i + 1}}}.

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.{{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}}

partition of an interval

Refinement of a partition

Norm of a partition

Applications

Tagged partitions

See also

References

Further reading