How to understand C(X)'' = bounded Borel measurable functions? Let $X$ be a compact metric space and $C(X)=\{ f:X\rightarrow \mathbb{R} \ | \ \ f \ continuous\}$ with the uniform norm. It is a separable Banach space.
1) I'm aware of the fact that $C(X)^*$, the space of continuous linear functionals $C(X)\rightarrow \mathbb{R}$ coincide with $M(X)$, the space of signed regular borel measures on $X$. 
This intuitively makes sense because $\mu\in M(X)$ can be naturally be seen as an evaluation map $f\mapsto \mathbb{R}$ by means of integration.
2) I'm also aware that $(C(X)^*)^*$, the dual of the dual, coincide with the set of bounded Borel measurable functions $F:X\rightarrow\mathbb{R}$ (source: P. Lax, Functional Analysis). 
UPDATE Landscape provided a counterexample in the comments.
My source was: P. Lax "Functional Analysis" 2002, Page 82, Theorem 14(ii).
I guess this might be a mistake, perhaps fixed in some errata somewhere.
New question:
C.f. Landscape's example, let $g_A$ be the extension (given by Hahn-Banach) of $f_A$ to $C([0,1])^{**}$. Looking at $g_A$ as a function $X\rightarrow\mathbb{R}$, $g_A$ must satisfy by construction the equation $\int_{[0,1]}g_A \ d \ \delta_x= 1$ if $x\in A$ and $0$ otherwise. This seems to imply that $g_A$ is the characteristic function of $A$. Hence not Borel if $A$ is not Borel.
Thus, $g_A$ as a function $(X\rightarrow\mathbb{R})$ is uniquely determined by the construction starting from $f_A$. But is it $g_A$ as an element of $C([0,1])^{**}$ uniquely determined? If so, it looks to me we would have a reasonably well defined notion of integration for non-measurable sets. 
Thanks!

Screen shot of Lax's theorem 14 on page 82:

 A: Let $X$ be a compact metric space.  Let $\mathcal B$ be the collection of Borel sets on $X$.  Banach space $C(X)$ is the set of all (necessarily bounded) continuous real-valued functions $f : X \to \mathbb R$, with norm
$$
\|f\|_\infty = \max\{|f(x)| : x \in X\}
$$
Banach space $M(X)$ is the set of all countably-additive signed ($\mathbb R$-valued) measures on the sigma-algebra $\mathcal B$.  The norm is the total variation: for $\mu \in M(X)$, write $\mu = \mu^+ - \mu^{-}$ where $\mu^+$ and $\mu^-$ are positive measures, singular to each other, and let
$$
\|\mu\|_1 = \mu^+(X)+\mu^-(X)
$$
The pairing $M(X) \times C(X) \to \mathbb R$ defined by
$$
\langle\mu,f\rangle = \int_X f \;d\mu
$$
identifies $C(X)^\ast$ isometrically with $M(X)$ in these norms.  
Another description of $M(X)$ may be obtained as follows.  Let $\{\tau_i : i \in I\}$ be a maximal (under inclusion) family of mutually singular probability measures on $(X,\mathcal B)$.  (Such a family exists by Zorn's Lemma.  But it is not unique.)  Identify $M(X)$ isometrically with the $l^1$-direct sum of the family of all the $L^1(\tau_i)$ spaces:
$$
M(X) \approx \left(\bigoplus_{i\in I} L^1(X,\mathcal B,\tau_i)\right)_{1}
$$
Once we describe $M(X)$ in this way, we get the corresponding description of the dual as the $l^\infty$-sum of the spaces $L^\infty(\tau_i)$.
$$
C(X)^{\ast\ast}\approx \left(\bigoplus_{i\in I} L^\infty(X,\mathcal B,\tau_i)\right)_{\infty}
$$
