Mesh of a partition

A mesh of a partition $P=\{x_0,\dots,x_n\}$ of $[a,b]$ is the number

For a tagged partition , the mesh depends only on the underlying partition $P$.

The mesh quantifies how “fine” a partition is; limits in the definitions of the Riemann integral and the Riemann–Stieltjes integral are taken as $\|P\|\to 0$.

Examples:

• The uniform partition of $[0,1]$ into $n$ equal pieces has mesh $\|P\|=1/n$.
• For $P=\{0,0.9,1\}$, the mesh is $\|P\|=0.9$.