Fine variation and fractal measures (Q1890624)

If \(\overline B_ r(x)\) is the closed ball centered at \(x\) and with radius \(r\) in an arbitrary metric space \((S, \rho)\) then the pair \((r, x)\) is said to be a constituent. A packing is a disjoint collection \(\pi\) of balls resp. constituents. If \(h\) is a finite, nonnegative constituent function, then the fine variation \(V_ *(h)\) of \(h\) is defined in two steps: 1. \(V_ \beta(h)= \sup \sum_{(r, x)\in \pi} h(r, x)\) for any covering \(\beta\) of the metric space \((S, \rho)\) and packing \(\pi\subset \beta\). 2. \(V_ *(h)= \inf_ \beta V_ \beta(h)\). The covering measure \({\mathcal C}^ s\) for some real \(s> 0\) due to St.- Raymond and Tricot is a variant of the Hausdorff measure \({\mathcal H}^ s\) following the device: replace open coverings by coverings with open balls centered at points within the covered set. If \(h(r, x)= f(x) (2r)^ s\) for \(f: S\to \mathbb{R}\) then \(V^ s_ *:= V_ *(h)\). The main result is that in the general metric setting \(V^ s_ *(\mathbf{1}_ E)= {\mathcal C}^ s(E)\) for \(E\subseteq S\). The author discusses also the changes necessary for this approach to Hausdorff as well as to spherical measures. This completes the discussion of the author starting with a recent other paper of the author dedicated to packing measures.