  In this paper, we construct new multifractal measures, on the Euclidean space
$\mathbb{R}^n$, in a similar manner to Hewitt-Stomberg measures but using the
class of all $n$-dimensional half-open binary cubes of covering sets in the
definition rather than the class of all balls. As an application we shall be
concerned with evaluation of Hewitt-Stromberg dimension of cartesian product
sets by means of the dimensions of their components.
