Stability and measurability of the modified lower dimension

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 ${\dim }_{ML}:\mathcal{K}(X)\to [0,\infty ]$ 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.

10 pages. A new paragraph added to the introduction

References in corpus (1)

• Lower Assouad Dimension of Measures and Regularity