Is Lebesgue outer measure semicontinous?

Let $m^*$ be the Lebesgue outer measure on $\mathbb{R}$. Endow $2^{[0,1]}$ with the product topology, which is Hausdorff compact. Identify it with the set of all subsets of $[0,1]$. Is the map $$m^*:2^{[0,1]}\to \mathbb{R}, \, A\mapsto m^*(A)$$lower semicontinuous? The question is inspired by Continuity from below for Lebesgue outer measure.