# Is the inner/outer measure mapping continuous?

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.

Let $$\Omega = \mathbb{N}$$ and let $$\mathcal{F}$$ be the field consisting of all finite and cofinite subsets of $$\mathbb{N}$$. Let $$\mu_n = \delta_{2n}$$ be a point mass at the integer $$2n$$, and let $$\mu$$ be the finitely additive measure that assigns measure $$0$$ to every finite set and $$1$$ to every cofinite set. Then $$\mu_n(F) \to \mu(F)$$ for every $$F \in \mathcal{F}$$.
Let $$A \subset \mathbb{N}$$ be the set of all even integers. Then we have $$\mu_n^i(A) = 1$$ for every $$n$$ but $$\mu^i(A) = 0$$.
• Thanks. On the other hand, if $\mathcal F$ is closed under arbitrary unions, then I think the result holds. I wonder if something like that is necessary, though. – aduh Mar 29 '20 at 23:30