Stability and measurability of the modified lower dimension

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.