On \(d\)-measure and \(d\)-dimension
A metric outer measure different from Hausdorff and packing measure is introduced as follows. For a bounded set \(F\subseteq\mathbb{R}^ d\) let \(N(F,r)\) be the minimum number of closed balls in \(\mathbb{R}^ d\) needed to cover \(F\). For \(0<\alpha\leq d\) define first \[ D^ \alpha(F)=\liminf_{r\to 0}N(F,r)r^ \alpha \] and then the metric outer measure \(d^ \alpha\) of \(E\subset\mathbb{R}^ d\) as \[ d^ \alpha(E)=\inf\left\{\sum^ \infty_{n=1}D^ \alpha(E_ n);\;E\subseteq\bigcup^ \infty_{n=1}E_ n,\;E_ n\quad\text{are bounded in }\mathbb{R}^ d\right\}. \] This gives the dimension index \(d-\dim\) in the usual way as for packing or Hausdorff measure. This measure construction is already contained in \textit{E. Hewitt} and \textit{K. Stromberg}: ``Real and abstract analysis'' (1965; Zbl 0137.032) as an exercise on page 145. The main result of this paper is concerned with the graph of functions \(f:[0,1]\to\mathbb{R}\) such that if for some \(\alpha\) with \(0<\alpha<1\) there exist positive constants \(C_ 1\) and \(C_ 2\) such that \(C_ 1| J|^ \alpha\leq\sup_{x,y\in J}| f(x)-f(y)|\leq C_ 2| J|^ \alpha\) for any interval \(J\) in \([0,1]\), then \(d\)-dimension and packing dimension are equal to \(2-\alpha\). This applies to the Weierstrass function, Kiesewetter curve and sums of Rademacher functions.
scientific article; zbMATH DE number 68377