# 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.

## Key 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.