Let $F$ be a finite set equipped the discrete topology. Let $X = F \times F \times ...$ be the countably infinite product space equipped with the product topology. Let $\mathcal A$ be any field of subsets of $X$ that contains the open subsets of $X$. Let $\mu$ be a finitely additive, finite measure with domain $\mathcal A$.

Suppose that $\mu$ has the following "clopen approximation property": For any $\epsilon > 0$ and any open subset $G$ of $X$, there is a clopen subset $C$ of $X$ such that $\mu(G \triangle C)< \epsilon$.

Is the clopen approximation property equivalent to the following "inner regularity property"? For every open subset $G$ of $X$, $\mu(G) = \sup\{\mu(C): C \subseteq G, C \text{ clopen}\}$.

Clearly inner regularity implies clopen approximation, but I am unable to see that the converse is true.

If $G$ is open, then it can be written as a countable union $G = C_1 \cup C_2 \cup ...$ of pairwise disjoint clopen sets. It seems reasonable to expect that if $G$ is approximable by *some* sequence of clopen sets, then it should be approximable from within by finite unions of its constiuent clopen sets. i.e. $\mu(\cup_{i=1}^n C_i) \to \mu(G)$ as $n \to \infty$. But, again, I am unable to see that this is the case.

If $G$ is approximable by some clopen sequence $B_n$, so that $\mu(G \triangle B_n) \to 0$, then $\mu(G - B_n) \to 0$ and $G - B_n$ is an *open* subset of $G$. If we could replace this sequence of open subsets of $G$ with a "similar" sequence of clopen subsets of $G$, maybe we could prove the result. I played around with sequences like $(C_1 \cup ... \cup C_n) - B_n$ (recall that $G = C_1 \cup C_2 \cup ...$), but didn't get anywhere.

The motivation for this has to do with finding countably additive extensions of finitely additive measures. The clopen approximation property can be used to characterize extreme points in the convex set of extensions from the clopen field to another, larger field. And, since the clopen sets form a compact class, inner regularity provides a sufficient condition for countable additivity. If the result in question holds, then I could say that the extreme points of the set of extensions from the clopen field to the field generated by open sets are countably additive on the latter.