Full packing dimensional projections of measures

We introduce a threshold parameter $D(\mu )$ for a Borel probability measure $\mu$ with compact support $E\subset {\mathbb{R}}^n$ such that, for every integer $1\le m\le n$, the orthogonal projection of $\mu$ onto a typical $m$ dimensional subspace attains full packing dimension if and only if $m\le D(\mu )$. In the complementary regime we show that the Assouad dimension of the support controls the possible drop of the packing dimension under projections:$${\dim }_P^m\mu \ge {\dim }_P\mu -\max \{0,{\dim }_{A}E-m\}\mathrm{.}$$ In particular, whenever $m\ge {\dim }_{A}E$, the packing dimension of every measure supported on $E$ is preserved under orthogonal projection onto almost every $m$-dimensional subspace. Taking supremum over the measures supported on a set recovers, in its Assouad dimension form, the corresponding result of Falconer, Fraser and Shmerkin for sets. A key ingredient, of independent interest, is a sharpening of an estimate of Falconer and Mattila for the growth of the measure of balls, in which the ambient dimension is replaced by the Assouad dimension of the support.