Partition of an interval explained

In mathematics, a partition of an interval on the real line is a finite sequence of real numbers such that

.

In other terms, a partition of a compact interval is a strictly increasing sequence of numbers (belonging to the interval itself) starting from the initial point of and arriving at the final point of .

Every interval of the form is referred to as a subinterval of the partition x.

Refinement of a partition

Another partition of the given interval [a, b] is defined as a refinement of the partition, if contains all the points of and possibly some other points as well; the partition is said to be “finer” than . Given two partitions, and, one can always form their common refinement, denoted, which consists of all the points of and, in increasing order.[1]

Norm of a partition

The norm (or mesh) of the partition

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

.

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 subject to the conditions that for each,

.

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.

Suppose that together with is a tagged partition of, and that together with is another tagged partition of . We say that together with is a refinement of a tagged partition together with if for each integer with, there is an integer such that and such that for some with . 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

Further reading

Notes and References

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