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.