A general mass transference principle

The Mass Transference Principle proved by Beresnevich and Velani (Ann. Math. (2) 164(3):971–992, 2006) is a celebrated and highly influential result which allows us to infer Hausdorff measure statements for lim sup sets of balls in Rⁿ from a priori weaker Lebesgue measure statements. The Mass Transference Principle and subsequent generalisations have had a profound impact on several areas of mathematics, especially Diophantine Approximation. In the present paper, we prove a considerably more general form of the Mass Transference Principle which extends known results of this type in several distinct directions. In particular, we establish a Mass Transference Principle for lim 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 Rⁿ. Furthermore, our main result is applicable in locally compact metric spaces and allows one to transfer Hausdorff g-measure statements to Hausdorff f-measure statements. We conclude the paper with an application of our mass transference principle to a general class of random lim sup sets.