A note on scaling properties of Hewitt Stromberg measure

Abstract

In this note, we investigate those Hewitt Stromberg measures which obey to a simple scaling law. Consider a dimension function $h$ and let $\HH^h$ be the corresponding Hewitt Stromberg measure. We say that $\HH^h$ obeys an order $\alpha$ scaling law whenever taking $A\subset \R^m$ and $c>0$, one has
$$
\HH^h(cA) = c^\alpha\HH^h(A).
$$