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 [xi, xi + 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 partitionP, 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 ∨ 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
x0 < x1 < x2 < … < xn
is the length of the longest of these subintervals[2][3]
A tagged partition[5] or Perron Partition 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 x0, …, xn together with t0, …, tn − 1 is a tagged partition of [a, b], and that y0, …, ym together with s0, …, sm − 1 is another tagged partition of [a, b]. We say that y0, …, ym together with s0, …, sm − 1 is a refinement of a tagged partitionx0, …, xn together with t0, …, tn − 1 if for each integeri with 0 ≤ i ≤ n, there is an integer r(i) such that xi = yr(i) and such that ti = sj for some j with r(i) ≤ j ≤ 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.