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The length of the longest subinterval in a partition.
A mesh of a partition of is the number
For a tagged partition, the mesh depends only on the underlying partition .
The mesh quantifies how “fine” a partition is; limits in the definitions of the Riemann integral and the Riemann–Stieltjes integral are taken as .
A tagged partition of is a partition together with a choice of points (tags) for each . It is often denoted .
Tagged partitions are the input data for a Riemann sum and, more generally, for a Riemann–Stieltjes sum.
A Riemann integral of a Riemann integrable function is the number
defined by
where and are the lower and upper sums over all partitions of .
A Riemann–Stieltjes integral of a bounded function with respect to an integrator function is a number
such that for every there exists with the property that, for every tagged partition with mesh ,
(The sum is called a Riemann–Stieltjes sum.)