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 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

x_{0} < x_{1} < x_{2} < … < x_{n}

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 t_{0}, …, t_{n − 1} subject to the conditions that for each i,

x_{i} ≤ t_{i} ≤ 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}, …, x_{n} together with t_{0}, …, t_{n − 1} is a tagged partition of [a, b], and that y_{0}, …, y_{m} together with s_{0}, …, s_{m − 1} is another tagged partition of [a, b]. We say that y_{0}, …, y_{m} together with s_{0}, …, s_{m − 1} is a refinement of a tagged partitionx_{0}, …, x_{n} together with t_{0}, …, t_{n − 1} if for each integeri with 0 ≤ i ≤ 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) ≤ 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.