Partition of an interval

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A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with one subinterval indicated in red.

In mathematics, a partition, P, of an interval [a, b] on the real line is a finite sequence of the form

a = x0 < x1 < x2 < ... < xn = b.

Every interval of the form [xi,xi+1] is referred to as a sub-interval.

Refinement of a partition[edit]

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.[1]

Norm of a partition[edit]

The norm (or mesh) of the partition

x0 < x1 < x2 < ... < xn

is the length of the longest of these subintervals,[2][3] that is

max{ |xixi−1| : i = 1, ..., n }.

Applications[edit]

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[edit]

A tagged partition[5] 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  x_0,\ldots,x_n together with  t_0,\ldots,t_{n-1} is a tagged partition of [a, b], and that  y_0,\ldots,y_m together with  s_0,\ldots,s_{m-1} is another tagged partition of [a,b]. We say that  y_0,\ldots,y_m and  s_0,\ldots,s_{m-1} together is a refinement of a tagged partition  x_0,\ldots,x_n together with  t_0,\ldots,t_{n-1} if for each integer i with  0 \le i \le n, there is an integer r(i) such that  x_i = y_{r(i)} and such that t_i = s_j for some j with  r(i) \le j \le 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[edit]

References[edit]

  1. ^ Brannan, D.A. (2006). A First Course in Mathematical Analysis. Cambridge University Press. p. 262. ISBN 9781139458955. 
  2. ^ Hijab, Omar (2011). Introduction to Calculus and Classical Analysis. Springer. p. 60. ISBN 9781441994882. 
  3. ^ Zorich, Vladimir A. (2004). Mathematical Analysis II. Springer. p. 108. ISBN 9783540406334. 
  4. ^ Limaye, Balmohan (2006). A Course in Calculus and Real Analysis. Springer. p. 213. ISBN 9780387364254. 
  5. ^ Dudley, Richard M. & Norvaiša, Rimas (2010). Concrete Functional Calculus. Springer. p. 2. ISBN 9781441969507. 

Further reading[edit]

  • 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.