  In this paper, we introduce a new notion called the \emph{box-counting
measure} of a metric space. We show that for a doubling metric space, an
Ahlfors regular measure is always a box-counting measure; consequently, if $E$
is a self-similar set satisfying the open set condition, then the Hausdorff
measure restricted to $E$ is a box-counting measure. We show two classes of
self-affine sets, the generalized Lalley-Gatzouras type self-affine sponges and
Bara\'nski carpets, always admit box-counting measures; this also provides a
very simple method to calculate the box-dimension of these fractals. Moreover,
among others, we show that if two doubling metric spaces admit box-counting
measures, then the multi-fractal spectra of the box-counting measures coincide
provided the two spaces are Lipschitz equivalent.
