mesh

The mesh of a partition is a single number measuring how coarse it is: the width of its widest strip. A small mesh means every piece is thin, so the approximating staircase cannot stray far from the curve anywhere. Driving the mesh to zero is the limiting process that defines the Riemann integral via tagged sums.

Formally, for a partition P = {x_0, ..., x_n} the mesh (also called the norm) is mesh(P) = max{ x_k - x_(k-1) : 1 <= k <= n }, the length of the largest subinterval. The Riemann integral is then the limit of Riemann sums as mesh(P) -> 0: for every epsilon > 0 there is delta > 0 so that every tagged partition with mesh < delta has its sum within epsilon of the integral.

Mesh and refinement capture different notions of fine. A refinement always has mesh no larger than the original, but small mesh does not imply refinement: two partitions can each have tiny mesh while sharing no interior points. The mesh-based definition (Riemann's) and the refinement-based definition (Darboux's) are equivalent for bounded functions, but the mesh formulation is what connects to numerical quadrature, where the mesh controls the error bound.