Let $\mathcal F$ be a field of subsets of a set $\Omega$. Equip the space $[0,1]^\mathcal F$ of functions from $\mathcal F$ into $[0,1]$ with the product topology. Then, the set $\Delta$ of finitely additive probability measures on $\mathcal F$ is a convex and compact subset of $[0,1]^\mathcal F$.

If $\mu \in \Delta$, define the inner measure $\mu^i: 2^\Omega \to [0,1]$ for $\mu$ by $$\mu^i(A) = \sup \big\{\mu(F): F \subset A, F \in \mathcal F \big\}, \ A \subset \Omega.$$ We can view the inner measure as a mapping $\mu \mapsto \mu^i$ from $\Delta$ into $[0,1]^{2^\Omega}$.

Question.Is the inner measure mapping continuous? That is, if $\mu_{\alpha}$ is a net in $\Delta$ that converges to $\mu$ (i.e. $\mu_\alpha(F) \to \mu(F)$ for all $F \in \mathcal F$), then is it true that $\mu^i_\alpha \to \mu^i$ (i.e. $\mu_\alpha^i(A) \to \mu^i(A)$ for every subset $A$ of $\Omega$)?

A similar question arises for outer measure, though I assume the answers are the same.