Cover of a set in the sense of Vitali

Question

I do not know how $\overline{D_\alpha}\geq \alpha$ makes ${\mathcal{F}}$ a Vitali cover of $D_\alpha$. Can someone explain to me why? Thank you.

Answer 1

There are two things you need to check: that $\mathcal{F}$ covers $E_\alpha$, and that it has the Vitali property.

To see that $\mathcal{F}$ covers $E_\alpha$, let $x\in E_\alpha$; then the upper derivative of $f$ at $x$ is $\ge\alpha$. This means I can find points $c$ and $d$ "near" $x$ such that the difference quotient ${f(d)-f(c)\over d-c}$ is "not much less than" $\alpha$; in particular, since $\alpha'<\alpha$, take $\epsilon={\alpha-\alpha'\over 2}$ and pick $c<x<d$ such that the difference quotient is at most $\epsilon$ below $\alpha$.

This shows that $\mathcal{F}$ covers $E_\alpha$. But indeed, if we examine the argument above, it's clear that $c$ and $d$ can be picked arbitrarily close to $x$ - the definition of limsup demands that the difference quotient be "large" arbitrarily close to $x$. So arbitrarily small intervals containing $x$ are in $\mathcal{F}$; so $\mathcal{F}$ is indeed a Vitali cover.