Partition of an interval

In mathematics, a partition of an interval $[a, b]$ on the real line is a finite sequence $x 0, x 1, x 2, \dots, x n$ of real numbers such that

a = x 0 < x 1 < x 2 < \dots < x n = 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 $[x i, x i + 1]$ 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 $P \lor Q$ , 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

x 0 < x 1 < x 2 < \dots < x n

is the length of the longest of these subintervals^[2]^[3]

max{| x i - x i -1 | : i = 1, \dots , 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 $t 0, \dots, t n - 1$ subject to the conditions that for each $i$ ,

x i \leq t i \leq 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. 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 $x 0, \dots, x n$ together with $t 0, \dots, t n - 1$ is a tagged partition of $[a, b]$ , and that $y 0, \dots, y m$ together with $s 0, \dots, s m - 1$ is another tagged partition of $[a, b]$ . We say that $y 0, \dots, y m$ together with $s 0, \dots, s m - 1$ is a refinement of a tagged partition $x 0, \dots, x n$ together with $t 0, \dots, t n - 1$ if for each integer $i$ with $0 \leq i \leq n$ , there is an integer $r (i)$ such that $x i = y r (i)$ and such that $t i = s j$ for some $j$ with $r (i) \leq j \leq r (i + 1) - 1$ . Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

References

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