The packing measure and symmetric derivation basis measure. II (Q1308849)

There is given a theorem which asserts that \(h^*_ s(E)= h_ s(E)= h_ p(E)\) for any subset \(E\) of \(\mathbb{R}^ n\) and any continuous increasing function \(h: \langle 0,\infty)\to \langle 0,\infty)\) with \(h(0)=0\). Let \(E\subset \mathbb{R}^ n\) and \(\{E_ k\}\) be its cover. The author calls a function \(\delta: \mathbb{R}^ n\to (0,\infty)\) a \(\delta^*\)-function of \(E\) and its cover \(\{E_ k\}\) whenever \(\delta^*(x)=\delta_ k>0\) iff \(x\in E_ k-(\bigcup_{i<k} E_ i)\) and \(\delta^*(x)=1\) iff \(x\not\in\bigcup_ i E_ i\). The author calls the number \(h^*_ s(E)\) the \(\delta^*\)-measure of \(E\) and \(h^*_ s(E)=\inf\{H^*_ s(E)\): for any \(\delta^*\)-function of \(E\) and its cover \(\{E_ k\}\}\), where \(H^*_ s(E)=\sup\{\sum_ i h(2r_ i): \{B(x_ i,r_ i)\}\) is a sequence of pairwise disjoint balls in \(\mathbb{R}^ n\) with \(x_ i\in E_ k-(\bigcup_{i<k} E_ i)\) for some \(k\) and \(r_ i<\delta_ k\}\). \(h_ s(E)\) is called the symmetric derivation basis measure of \(E\) and \(h_ s(E)=\inf\{\sup\{\sum_ i h(2r_ i): \{B(x_ i,r_ i)\}\) is a sequence of pairwise disjoint balls in \(\mathbb{R}^ n\) with \(x_ i\in E\) and \(r_ i<\delta(x_ i)\}\); \(\delta: \mathbb{R}^ n\to (0,\infty)\}\). The number \(h_ p(E)\) is called packing measure of \(E\) and it holds: \(h_ p(E)=\inf\{\sum_ i H_ p(E_ i): E\subset\bigcup_ i E_ i\}\), where \(H_ p(A)=\inf_{\delta\to 0}\{\sup\{\sum_ i h(2r_ i): \{B(x_ i,r_ i)\}\) is a sequence of pairwise disjoint balls in \(\mathbb{R}^ n\) with \(x_ i\in A\) and \(r_ i<\delta\}\}\). [Cf. also Part I, same journal 17, No. 1, 421-422 (1992; Zbl 0749.28005)].