An Exact Formula for the Lower Hewitt-Stromberg Measure of Self-Similar Sets and Its Continuity

Abstract

This manuscript investigates the spherical density of the lower Hewitt–Stromberg measure in \(\mathbb {R}^d\). Also, we establish that if \(S=(S_1,\ldots ,S_n)\) is an iterated function system fulfilling the strong open set condition for some open set U, then

$$\begin{aligned} \mathcal {U}^\alpha (K_S \cap U(x,r)) \le (2r)^\alpha , \end{aligned}$$

for every open ball \(U(x,r)\subset U\) with \(x\in K_S\) and \(r>0\). Employing this estimate, we derive an exact formula for the lower Hewitt–Stromberg measure of self-similar sets. As a consequence, we show that the mapping

$$\begin{aligned} M_{SSC} \longrightarrow \mathbb {R}: S \longmapsto \mathcal {U}^\alpha (K_S), \end{aligned}$$

is continuous, where \(M_{SSC}\) denotes the class of all iterated function systems fulfilling the strong separation condition.