Partition of an interval



In mathematics, a partition of an interval $[a, b]$ on the real line is a finite sequence $x_{0}, x_{1}, x_{2}, …, x_{n}$ of real numbers such that



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 $a = x_{0} < x_{1} < x_{2} < … < x_{n} = b$ 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 $[x_{i}, x_{i + 1}]$, which consists of all the points of $P$ and $Q$, in increasing order.

Norm of a partition
The norm (or mesh) of the partition

is the length of the longest of these subintervals

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.

Tagged partitions
A tagged partition or Perron Partition is a partition of a given interval together with a finite sequence of numbers $P ∨ Q$ subject to the conditions that for each $i$,

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 $x_{0} < x_{1} < x_{2} < … < x_{n}$ together with $max{|x_{i} − x_{i−1}| : i = 1, …, n }$ is a tagged partition of $t_{0}, …, t_{n − 1}$, and that $x_{i} ≤ t_{i} ≤ x_{i + 1}$ together with $x_{0}, …, x_{n}$ is another tagged partition of $t_{0}, …, t_{n − 1}$. We say that $[a, b]$ together with $y_{0}, …, y_{m}$ is a refinement of a tagged partition $s_{0}, …, s_{m − 1}$ together with $[a, b]$ if for each integer $i$ with $y_{0}, …, y_{m}$, there is an integer $s_{0}, …, s_{m − 1}$ such that $x_{0}, …, x_{n}$ and such that $t_{0}, …, t_{n − 1}$ for some $j$ with $0 ≤ i ≤ n$. Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.