Definition

Let $f:[0,1]\to \mathbb{R}$, let $0=t_{0}<t_{1}<\dots<t_{N}=1$ be a subdivision of $[0,1]$ where $\pi=(t_{0},\dots,t_{N})$. We define the mesh of our subdivision $\pi$ or the fineness of of our subdivision to be $$\delta(\pi)=\max_{i=0,\dots ,N-1}|t_{i+1}-t_{i}|$$