Some Remarks on Hausdorff Measurability of Lipschitz Images in Metric Spaces

TL;DR

This paper proves that Lipschitz images of certain subsets in metric spaces are Hausdorff-measurable, linking concepts across analysis and topology.

Contribution

It establishes Hausdorff measurability of Lipschitz images of subsets with Hausdorff-null boundaries in metric spaces, extending understanding in measure theory.

Findings

Lipschitz open-map images of subsets with Hausdorff-null boundary are Hausdorff-measurable.

Results connect measure theory with complex analysis, functional analysis, and topology.

Provides conditions under which Lipschitz images are measurable in metric spaces.

Abstract

In this short note, we show that, in any given metric space, every Lipschitz open-map image of every subset of a given metric space whose boundary is Hausdorff-null is Hausdorff-measurable with respect to the same dimension. The main results are connected with a number of familiar concepts in other branches such as complex analysis, functional analysis, and topology.