Let $K$ be a compact Hausdorff space. We let $Z$ be the closed subspace generated by $\{\chi\_{F}:F$ closed $G\_{\delta}$ sets in $K\}$ in $C(K)^{\*\*}$. My question is the following:


Question 1. Is $C(K)\subseteq Z\subseteq C(K)^{\*\*}$?


Since each $f\in C(K)$ is Baire measurable, each such $f$ is the unifrom limit of a sequence of simple Baire measurable functions. But I want a stronger result:


Question 2. For $f\in C(K)$ and every $\epsilon>0$, does there exist $g=\sum\limits\_{k=1}^{n}a\_{k}\chi\_{F\_{k}}$ (where each $F\_{k}$ closed $G\_{\delta}$ sets in $K)$ such that $|f(t)-g(t)|<\epsilon$ for all $t\in K$?



A subspace generated by closed $G_{\delta}$ sets in $K$ between $C(K)$ and $C(K)^{**}$
Question 2 has a positive answer. For any $f \in C(K)$, let $a = \inf\_{t\in K} f(t)$ and let $b = \sup\_{t\in K} f(t)$. For any $n \in \mathbb{N}$ with $n > 0$, we can set $$ g\_n = a + \sum\_{k=1}^n \frac{b-a}{n} \chi\_{F\_k},$$ where $ F\_k = \{t \in K : f(t) \geq a + \frac{k}{n}(b-a)\} $, which is a closed $G\_\delta$ set ($F\_k = \bigcap\_{\ell \in \mathbb{N}}\{t \in K : f(t) > a + \frac{k}{n}(b-a) - \frac{1}{\ell + 1}\}$). Since the constant $a$ can be thought of as $a\chi\_K$, $g\_n$ is of the required form. By construction, $|f(t) - g(t)| \leq \frac{b-a}{n}$ for every $t\in K$, which we can clearly make arbitrarily small.


