Vitali covers and fine covers on the real line by closed intervals

Let $E\subseteq \mathbb{R}$, and let $\mathcal{V}$ be a family of bounded closed nondegenerate intervals Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, so every member has the form $[u,v]$ with $u<v$.

We say that $\mathcal{V}$ is a fine cover of $E$ when for every $x\in E$ and every $\delta >0$ there is an interval $I\in \mathcal{V}$ such that $x\in I$ and $\mathrm{\mid }I\mathrm{\mid }<\delta$, where $\mathrm{\mid }[u,v]\mathrm{\mid }:=v-u$.

On this page, a Vitali cover of $E$ means the same thing: a cover by bounded closed intervals that shrinks to each covered point. In one dimension this is the interval form of the ball-language statement.

• The word "fine" emphasizes the shrinking property. An arbitrary interval cover of $E$ need not be fine.
• Closed intervals are used because that is the exact one-dimensional form proved later on this page; replacing them by balls gives the same theorem on the line.