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 [Adv. Math. 329 (2018), pp. 273–328] introduced the modified lower dimension $dim_\textit {{ML}}$ by making the lower dimension monotonic with the simple formula $dim_\textit {{ML}}X=\sup \{\dim _L E: E\subset X\}$.

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 $dim_\textit {{ML}}\colon \mathcal {K}(X)\to [0,\infty ]$ 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.