Partition of an interval
In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x0, x1, x2, …, xn of real numbers such that
- a = x0 < x1 < x2 < … < xn = b.
In other terms, a partition of a compact interval I is a strictly increasing sequence of numbers (belonging to the interval I itself) starting from the initial point of I and arriving at the final point of I.
Every interval of the form [xi, xi + 1] is referred to as a subinterval of the partition x.
Refinement of a partition
Another partition Q of the given interval [a, b] is defined as a refinement of the partition P, if Q 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, in increasing order.[1]
Norm of a partition
The norm (or mesh) of the partition
- x0 < x1 < x2 < … < xn
is the length of the longest of these subintervals[2][3]
- max{|xi − xi−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.[4]
Tagged partitions
A tagged partition[5] or Perron Partition is a partition of a given interval together with a finite sequence of numbers t0, …, tn − 1 subject to the conditions that for each i,
- xi ≤ ti ≤ xi + 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. 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.[citation needed]
Suppose that x0, …, xn together with t0, …, tn − 1 is a tagged partition of [a, b], and that y0, …, ym together with s0, …, sm − 1 is another tagged partition of [a, b]. We say that y0, …, ym together with s0, …, sm − 1 is a refinement of a tagged partition x0, …, xn together with t0, …, tn − 1 if for each integer i with 0 ≤ i ≤ n, there is an integer r(i) such that xi = yr(i) and such that ti = sj for some j with r(i) ≤ j ≤ r(i + 1) − 1. 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
References
- ↑ Brannan, D. A. (2006). A First Course in Mathematical Analysis. Cambridge University Press. p. 262. ISBN 9781139458955. https://books.google.com/books?id=N8bL9lQUGJgC&pg=PA262.
- ↑ Hijab, Omar (2011). Introduction to Calculus and Classical Analysis. Springer. p. 60. ISBN 9781441994882. https://books.google.com/books?id=_gb9fMqur9kC&pg=PA60.
- ↑ Zorich, Vladimir A. (2004). Mathematical Analysis II. Springer. p. 108. ISBN 9783540406334. https://books.google.com/books?id=XF8W9W-eyrgC&pg=PA108.
- ↑ Ghorpade, Sudhir; Limaye, Balmohan (2006). A Course in Calculus and Real Analysis. Springer. p. 213. ISBN 9780387364254. https://books.google.com/books?id=Ou53zXSBdocC&pg=PA213.
- ↑ Dudley, Richard M.; Norvaiša, Rimas (2010). Concrete Functional Calculus. Springer. p. 2. ISBN 9781441969507. https://books.google.com/books?id=fuuB59EiIagC&pg=PA2.
Further reading
- Gordon, Russell A. (1994). The integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics, 4. Providence, RI: American Mathematical Society. ISBN 0-8218-3805-9.
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