Yet a shorter proof of an inequality of Cutler and Olsen (Q5932463)

A new proof is given of the inequality NEWLINE\[NEWLINE \dim E\leq \sup _{\mu}\mathop{\underline \lim}\limits _{\delta \to 0+}\inf \Biggl\{ \sum _{i\in \mathbf N}\mu E_i \frac {\log \mu E_i}{\log \delta}: \{E_i\}\text{ is a disjoint \(\delta \)-cover of \(E\)}\Biggr\}, NEWLINE\]NEWLINE where \(\dim E\) is the Hausdorff dimension of a Borel set \(E\subset \mathbf R^n\) and \(\mu \) runs over the family of all Borel probability measures on \(E\).