Mesh of a partition

Let $P$ be a partition of $[a,b]$ given by $a=x_0<\cdots<x_n=b$. The mesh (or norm) of $P$ is

The mesh measures how fine a partition is. Many convergence statements for Riemann sums are phrased in terms of $\|P\|\to 0$.

Examples:

• For the uniform partition $x_i=a+i(b-a)/n$, one has $\|P\|=(b-a)/n$.
• If $P:0<0.9<1$, then $\|P\|=0.9$.
• Refining a partition cannot increase the mesh: if $Q$ refines $P$, then $\|Q\|\le \|P\|$.