  In this paper we prove an upper bound on the "size" of the set of
multiplicatively $\psi$-approximable points in $\mathbb R^d$ for $d>1$ in terms
of $f$-dimensional Hausdorff measure. This upper bound exactly complements the
known lower bound, providing a "zero-full" law which relates the Hausdorff
measure to the convergence/divergence of a certain series in both the
homogeneous and inhomogeneous settings. This zero-full law resolves a question
posed by Beresnevich and Velani (2015) regarding the "log factor" discrepancy
in the convergent/divergent sum conditions of their theorem. We further prove
the analogous result for the multiplicative doubly metric setup.
