Stability and measurability of the modified lower dimension

TL;DR

This paper investigates the properties of the modified lower dimension, proving its stability, providing a new characterization, and analyzing its measurability, thereby answering several open questions in the field.

Contribution

It establishes the finite stability of the modified lower dimension, offers a new characterization, and analyzes its Borel measurability properties.

Findings

Modified lower dimension is finitely stable in any metric space.

The map im_{ML} is Borel measurable of Baire class 2.

It is not Borel measurable on the space of closed sets in ll^1.

Abstract

The lower dimension $\dim_L$ is the dual concept of the Assouad dimension. As it fails to be monotonic, Fraser and Yu introduced the modified lower dimension $\dim_{ML}$ by making the lower dimension monotonic with the simple formula $\dim_{ML} X=\sup\{\dim_L E: E\subset X\}$. As our first result we prove that the modified lower dimension is finitely stable in any metric space, answering a question of Fraser and Yu. We prove a new, simple characterization for the modified lower dimension. For a metric space $X$ let $\mathcal{K}(X)$ denote the metric space of the non-empty compact subsets of $X$ endowed with the Hausdorff metric. As an application of our characterization, we show that the map $\dim_{ML} \colon \mathcal{K}(X)\to [0,\infty]$ is Borel measurable. More precisely, it is of Baire class $2$, but in general not of Baire class $1$. This answers another question of Fraser and Yu. Finally, we prove that the modified lower dimension is not Borel measurable defined on the closed sets of $\ell^1$ endowed with the Effros Borel structure.