If the finitely additive measure of an open set is approximable by clopen sets, is it approximable from within?

Question

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.

Answer 1

The answer to this question is negative. To construct a counterexample, fix any free ultrafilter $\mathcal U$ on $X$ containing a discrete subspace $D$ of $X$. The characteristic function $u:\mathcal P(X)\to \{0,1\}$ of this ultrafilter determines a $\{0,1\}$-valued finitely additive measure $u$ defined on the algebra $\mathcal P(X)$ of all subsets of $X$.

We claim that for every subset $S\subset X$ there exists a clopen set $C\subset X$ with $u(C\triangle S)=0$. Indeed, if $u(S)=0$, then for the clopen set $C=\emptyset$ we have $u(C\triangle S)=u(S)=0$. In $u(S)=1$, then for the clopen set $C=X$ we have $u(C\triangle S)=u(X\setminus S)=u(X)-u(S)=1-1=0$.

Let $A$ be the (closed) set of accumulation points of the discrete subspace $D$ in the compact metrizable space $X$. The open set $U=X\setminus A$ contains the set $D$ and hence has measure $u(U)=1$. On the other hand, for any clopen set $K\subset U$, the set $K$ is compact and disjoint with $A$. Then the intersection $D\cap K$ is finite and hence $K\notin \mathcal U$, which means that $u(K)=0$. Therefore, $$u(U)=1\ne 0=\sup\{u(K):K\subset U\mbox{ is a clopen subset of $X$}\}.$$

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Now we shall modify the measure $u$ in order to make it strictly positive.

Let $p$ be the standard product probability measure defined on the $\sigma$-algebra $\mathcal B(X)$ of Borel subsets of $X$. It is clear that $p$ is strictly positive in the sense that $p(U)>0$ for any non-empty open subset $U\subset X$.

Consider the strictly positive probability measure $\mu=\frac12u+\frac12p$, defined on the $\sigma$-algebra $\mathcal B(X)$.

We can chose the discrete subset $D$ so that its closure $\bar D=A\cup D$ has product measure $p(\bar D)=0$. Then the open set $X\setminus A$ has measure $$\mu(X\setminus A)=1\ge \frac{1}{2}=\sup\{\mu(U):U\subset X\setminus A\mbox{ is a clopen set in $X$}\}.$$

Now we prove that for any Borel subset $B\subset X$ and every $\varepsilon>0$ there exists a clopen subset $C\subset X$ such that $\mu(B\triangle C)<\varepsilon$.

If $u(B)=0$, then by the regularity of the measure $p$, find a compact subset $K\subset B\setminus A$ such that $p(K)>p(B)-\varepsilon$. Using the regularity of the measure $p$ once more, find a clopen subset $C\subset X$ such that $K\subset C\subset X\setminus A$ and $p(C)<p(K)+\varepsilon$. Then $$p(C\triangle B)\le p(C\triangle K)+p(K\triangle B)<\varepsilon+\varepsilon=2\varepsilon$$ and $$\mu(C\triangle B)=\tfrac12p(C\triangle B)+\tfrac12u(C\triangle B)<\tfrac122\varepsilon+\tfrac12u(C)+\tfrac12u(B)=\varepsilon+0+0=\varepsilon.$$

If $u(B)=1$, then by the regularity of the measure $p$, there exists a compact subset $K\subset B$ such that $A\subset K$ and $p(K)>p(B)-\varepsilon$. Using the regularity of the measure $p$ once more, find a clopen subset $C\subset X$ such that $K\subset C$ and $p(C)<p(K)+\varepsilon$. Then $$p(C\triangle B)\le p(C\triangle K)+p(K\triangle B)<\varepsilon+\varepsilon=2\varepsilon.$$ Since $C$ is a neighborhood of $A$, the set $D\setminus C$ is finite and hence $u(C\cap D)=u(D)-u(D\setminus C)=1-0=1$. Since $\mathcal U$ is a filter, $C\cap B=1$ and hence $u(C\cap B)=1$ and $u(C\triangle B)=0$. Then $$\mu(C\triangle B)=\tfrac12p(C\triangle B)+\tfrac12u(C\triangle B)<\tfrac122\varepsilon+0=\varepsilon.$$