A General Mass Transference Principle

TL;DR

This paper introduces a broad generalization of the Mass Transference Principle, enabling the transfer of measure statements for a wide class of sets including self-similar sets and manifolds, in various metric spaces.

Contribution

It extends the Mass Transference Principle to more general settings and types of sets, broadening its applicability in geometric measure theory.

Findings

Proves a general form of the Mass Transference Principle for $ ext{limsup}$ sets.

Applicable to self-similar sets and smooth manifolds in $ extbf{R}^n$.

Extends previous results to locally compact metric spaces.

Abstract

In this paper we prove a general form of the Mass Transference Principle for $\limsup$ 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.