Stability and measurability of the modified lower dimension

Abstract

The lower dimension dim L \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 d i m {ML} dim_\textit {{ML}} by making the lower dimension monotonic with the simple formula d i m {ML} X = sup { dim L ⁡ E : E ⊂ X } 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 X let K ( X ) \mathcal {K}(X) denote the metric space of the non-empty compact subsets of X X endowed with the Hausdorff metric. As an application of our characterization, we show that the map d i m {ML} : K ( X ) → [ 0 , ∞ ] dim_\textit {{ML}}\colon \mathcal {K}(X)\to [0,\infty ] is Borel measurable. More precisely, it is of Baire class 2 2 , but in general not of Baire class 1 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 ℓ 1 \ell ^1 endowed with the Effros Borel structure.