Stability and measurability of the modified lower dimension

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