Double dual of the space $C[0,1]$

Question

The second dual or double dual of the space of all continuous functions on $[0,1]$, $C[0,1]$ is von Neumann algebra. Can anyone help me identifying this space?

Answer 1

The following just regurgitates the links in David Ullrich's answer, but Ullrich doesn't give them the emphasis they deserve.

Let $S$ be a maximal set of mutually singular measures on $[0,1]$; such a set always exists by Zorn's Lemma. Then, for any $\mu\in C([0,1])^*$, we have $$\mu=\sum_{s\in S; s\ll\mu}{\frac{d\mu}{ds}s}$$ Moreover, by singularity, $$\|\mu\|_{C([0,1])^*}=\sum_{s\in S;s\ll\mu}{\left\|\frac{d\mu}{ds}\right\|_{L^1(s)}\|s\|_{C([0,1])^*}}$$ which shows that $C([0,1])^*$ is isometric to the $l^1$ direct sum $$C([0,1])^*\cong{\bigoplus_{s\in S}}^{l^1}{L^1(s)}\cong L^1\left({\oplus S}\right)$$ where $\oplus S$ denotes the measure on $(S;\text{discrete})\times[0,1]$ given by $$(\oplus S)(A)=\sum_{s\in S}{s(\{x:(s,x)\in A\})}$$ (That is, take $S$-many copies of $[0,1]$, each with their own natural measure. Conjoin them together into an incredibly long line.)

Since $(L^1)^*=L^{\infty}$, $$C([0,1])^{**}\cong L^{\infty}(\oplus S)$$

^{(Of course, whether you can find a useful description of $S$ is a separate question.)}