Let $\alpha$ denote the Kuratowski measure of noncompactness of a set, and $X$ an infinite dimensional Banach space. If $M_1,M_2,...$ are countably many bounded subsets of $X$ with finite measures $\alpha(M_j)$, and if $M_{j}\subseteq M_{j+1}$ for all $j$, then is it the case that
$$ \alpha \left(\bigcup _{j=1}^{\infty }M_{i}\right)=\lim _{j\to \infty }\alpha (M_{j})\,\,?$$
I know that "measures" satisfy this property, but as I learned it measures are countably additive, while $\alpha$ is countably sub-additive, so it is not what we usually call a measure.
Thanks for your help! I apologize if this is trivial; I haven't taken topology.
Edit: the Kuratowski measure: https://en.wikipedia.org/wiki/Measure_of_non-compactness