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

a = x₀ < x₁ < x₂ < ... < x_n = b.

Such partitions are used in the theory of the Riemann integral and the Riemann-Stieltjes integral.

The norm (or mesh) of the partition

x₀ < x₁ < x₂ < ... < x_n

is the length of the longest of these subintervals; it is

max{ |x_i − x_i−1| : i = 1, ..., n }.

As the mesh approaches zero, a Riemann sum based on the partition approaches the Riemann integral.

A tagged partition is a partition of an interval together with a finite sequence of numbers t_{0, ..., tn−1 subject to the conditions that for each i,}

x_{i $\backslash le$ ti $\backslash le$ xi+1.}

In other words, it is a partition together with a distinguished point of every subinterval. The mesh of a tagged partition is defined the same as for an ordinary partition. We can 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 $x\_0,\backslash ldots,x\_n$ together with $t\_0,\backslash ldots,t\_\{n-1\}$ are a tagged partition of $b$, and that $y\_0,\backslash ldots,y\_m$ together with $s\_0,\backslash ldots,s\_\{m-1\}$ are another tagged partition of $*$. We say that $y\_0,\backslash ldots,y\_m$ and $s\_0,\backslash ldots,s\_\{m-1\}$ together are a refinement of $x\_0,\backslash ldots,x\_n$ together with $t\_0,\backslash ldots,t\_\{n-1\}$ if for each integer $i$ with $0\; \backslash le\; i\; \backslash le\; n$, there is an integer $r(i)$ such that $x\_i\; =\; y\_\{r(i)\}$ and such that $t\_i\; =\; s\_j$ for some $j$ with $r(i)\; \backslash le\; j\; \backslash le\; r(i+1)$. Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

Mathematical analysis

Partición de un intervalo | Partizione di un intervallo | Skaidinys | Välin jako