Existence of a probability measure under which a subset of $[0,1]$ cannot be approximated by a Borel set

Question

Is there some probability measure, $p$, on $\left(\left[0,1\right],2^{\left[0,1\right]}\right)$ relative to which some $D\subseteq\left[0,1\right]$ cannot be approximated by a Borel set, i.e. if $E$ is any Borel set $E\in\mathfrak{B}\left(\left[0,1\right]\right)$, $p\left(D\Delta E\right)>0$?

Answer 1

Yes. Assuming there is a real valued measurable cardinal, this will be true of any total extension $p$ of Lebesgue measure on $[0, 1]$. This is because the measure algebra of any such extension is not separable. This fact is due to Gitik and Shelah.

If you want to directly construct such a measure, you can start with a model of $GCH$ with a measurable cardinal $\kappa$ and add $\kappa$ random reals. In the resulting model (due to Solovay), $2^{\omega} = \kappa$ and one can easily construct a total extension $p$ of Lebesgue measure such that the measure algebra of $p$ is isomorphic to random forcing for adding $(2^{\omega})^{+}$ many random reals.

Edit: I think the following also holds: Let $p$ be any total measure on $[0, 1]$ that vanishes on singletons. Then there is a set $X$ such that $p(X \cap E) = p(X^c \cap E) = p(E)/2$ for every Borel set $E$. This follows from the result of Gitik Shelah mentioned above combined with a theorem of Maharam.