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

The Banach-space dual of $C[0, 1]$ is $M[0, 1]$, the space of complex Borel measures on $[0, 1]$. The dual of $M[0, 1]$ is the “enveloping von Neumann algebra” of $C[0, 1]$ and is fairly intractable. However, assuming the continuum hypothesis, R. D. Mauldin 1 proved the following representation theorem for bounded linear functionals on $M[0, 1]$: for every such functional $T$ there is a bounded function from the set of Borel subsets of $[0, 1]$ to $\mathbb{C}$ such that $$ T(\mu)=\int\psi\, d\mu $$ for all $\mu\in M[0,1]$. Here the integral notation signifies the limit over the directed set of Borel partitions $\{B_1,\ldots,B_n\}$ of $[0,1]$ of the quantity $\sum\psi(B_i)\mu(B_i)$, and part of the assertion is that $\psi$ can be chosen so that this limit always exists. The proof begins by choosing a maximal family of mutually singular Borel measures on $[0,1]$; using the continuum hypothesis, this family is then indexed by countable ordinals and $\psi$ is then defined by transfinite recursion.

1 R. D. Mauldin, A representation theorem for the second dual of $C[0, 1]$, Studia Math., 46 (1973), 197–200.

I recently saw it asserted that the second dual of $C[0,1]$ was $\mathcal L^\infty[0,1]$, the space of bounded Borel functions. I was surprised at this, tried to prove it, couldn't quite make the details work. Finally saw a simple proof it was not so. (See the Amusing Note below regarding what $\mathcal L^\infty[0,1]$ actually is.) Started to post something about that, this thread popped up as possibly related.

Here, here and here we see much more complete discussions of that second dual; it seems perhaps nonetheless worthwhile to post the counterexample here, just for the sake of giving a very elementary proof that it's not as simple as one might possibly think.

Motivation: A few approaches to proving the false result would have worked if only I could show that the function $x\mapsto\Lambda\delta_x$ was Borel. Finding an example where that was not so turned out to give a counterexample to the result itself:

Say $\delta_x$ is a point mass at $x$, and let $X$ be the span of $\delta_x$ for $x\in[0,1]$. Suppose $f:[0,1]\to\Bbb C$, $|f(t)|\le1$ for all $t$, and $f$ is not Borel. Define $\Lambda:X\to\Bbb C$ by $$\Lambda\left(\sum_{j=1}^nc_j\delta_{x_j}\right)=\sum_{j=1}^nc_jf(x_j)$$(or $\Lambda\mu=\int f\,d\mu$). Since $||\sum c_j\delta_{x_j}||=\sum|c_j|$ it follows that $||\Lambda||\le 1$; now Hahn-Banach extends $\Lambda$ to $\overline\Lambda\in\mathcal M[0,1]^*$. And $\overline\Lambda$ is not given by integration against $g\in\mathcal L^\infty$. Because if it were then $$g(t)=\int g\,d\delta_t=\overline\Lambda \delta_t=\Lambda\delta_t=f(t),$$but $f$ is not Borel.

In fact in retrospect it's clear that the norm-closed span of the $\delta_x$ is just $\ell^1[0,1]$, with dual $\ell^\infty[0,1]$. Already much larger than $\mathcal L^\infty[0,1]$, and of course an element of the second dual of $C[0,1]$ is not determined by its action on the $\delta_x$.

Amusing Note Of course $C[0,1]$ is weak* dense in the second dual. In fact $\mathcal L^\infty[0,1]$ is precisely the closure of $C[0,1]$ in the second dual under sequential weak* convergence. Hint: This is the same as showing that $\mathcal L^\infty[0,1]$ is the closure of $C[0,1]$ under "uniformly bounded sequential pointwise" convergence. (And note that of course you don't get the closure of a set under a sequential limit operation by adding all the sequential limits; in fact you have to repeat "add all the sequential limits" $\omega_1$ times.)