Differentiability, Porosity and Doubling in Metric Measure Spaces

TL;DR

This paper explores the relationship between differentiable structures and measure properties in metric measure spaces, showing that differentiability implies measure zero porous sets and pointwise doubling, but approximate differentiability does not guarantee doubling.

Contribution

It establishes a link between differentiability and measure doubling, and provides a construction demonstrating the limits of approximate differentiability.

Findings

Porous sets have measure zero in spaces with differentiable structures.

Differentiability implies measure is pointwise doubling.

Approximate differentiability does not ensure measure doubling.

Abstract

We show if a metric measure space admits a differentiable structure then porous sets have measure zero and hence the measure is pointwise doubling. We then give a construction to show if we only require an approximate differentiable structure the measure need no longer be pointwise doubling.