A generalization of Marstrand's theorem

TL;DR

This paper extends Marstrand's projection theorem to general separable metric spaces under broad conditions, providing a flexible and simplified approach that encompasses many classical results in geometric measure theory.

Contribution

It introduces a generalization of Marstrand's theorem applicable to separable metric spaces with minimal assumptions, using simpler proof techniques than traditional methods.

Findings

Generalized Marstrand's theorem for separable metric spaces

Proofs rely on measurable projections and transversality hypotheses

Framework recovers most classical Marstrand-like theorems

Abstract

In this paper we prove two general results related to Marstrand's projection theorem in a quite general formulation over separable metric spaces under a suitable transversality hypothesis (the "projections" are in principle only measurable) - the result is flexible enough to, in particular, recover most of the classical Marstrand-like theorems. Our proofs use simpler tools than many classical works in the subject, where some techniques from harmonic analysis or special geometrical structures on the spaces are used.