On the second dual of $C[0,1]$ I have two questions on the second dual of  $C[0,1]$:
R. D. Mauldin ([1]) proved that: For a given bounded linear functional 
$T: C[0,1]^*\to \mathbb{C}$ there is a bounded function $\psi$ defined on $B$, the set of all Borel subsets of $[0,1]$, with $T(\mu)=\int \psi d\mu$.
I guess the following two items. Could you please let me know your ideas about?
1) Let $\psi$ be a bounded function on $B$ which is also in $C[0,1]^{**}$. The restriction $\psi_{|_{X}}$ forms a bounded Borel measurable function.   
2) We denote $V_{\infty}[0,1]$, by the set of all bounded Borel measurable functions on $[0,1]$. Then $\psi\to \psi_{|_{X}}$ defines a * homomorphism from 
 $C[0,1]^{**}$ onto  $V_{\infty}[0,1]$. It implies that  $V_{\infty}[0,1]=\frac{C[0,1]^{**}}{J}$ where $J$ is a closed ideal. 
 A: It is easy to see that $\int \psi d\delta_t$ where $\delta_t$ is the Dirac mass at $t$ is $\psi(\{t\})$
So you are starting from a $T \in C([0,1])^{**}$ and you are attaching to it the function $t \mapsto T(\delta_t)$. This has absolutely nothing to do with Mauldin's result, so I'm not sure why you are mentioning it.
Now let $V \subset C([0,1])^*$ be the vector subspace spanned by dirac masses.
It is easy to see that $v = \sum a_i \delta_{x_i}$ is a finite linear combination of (distinct) Dirac masses, then $\Vert v \Vert = \sum |a_i|$.
Hence, pick $f$ any bounded function on $X$, the linear map that send $\delta_{x_i}$ to $f(x_i)$ is continuous on $V$, of norm $\max |f(x)|$.
By the Hahn-Banach theorem it extend into an element of $C([0,1])^{**}$ of the same norm. 
So this means that $\psi_{|X}$ can be basically any bounded function on $[0,1]$. In particular it has no reason to be measurable.
For you second question, if you replace $V$ by the set of all bounded functions on $[0,1]$ then the answer is yes:
$V$ is a von neumann algebra and there is a morphism $C[0,1] \rightarrow V$ so this can be extended into a normal morphism $C[0,1]^{**} \rightarrow V$. this extension indeed corresponds to the map described above hence is a surjective normal morphism of von Neuman algebra, hence $V$ is a quotient of $C[0,1]^{**}$, by a two sideed ideal, but even by a projection !
