A General Mass Transference Principle

In this paper we prove a general form of the Mass Transference Principle for $\mathrm{lim\hspace{0.17em}sup}$ sets defined via neighbourhoods of sets satisfying a certain local scaling property. Such sets include self-similar sets satisfying the open set condition and smooth compact manifolds embedded in ${\mathbb{R}}^n$. Our main result is applicable in locally compact metric spaces and allows one to transfer Hausdorff $g$-measure statements to Hausdorff $f$-measure statements. This work extends previous results of this type in several distinct directions.