Solovay [shows](http://www.ams.org/books/pspum/013.1/0290961/pspum013.1-0290961.pdf) that the existence of a measurable cardinal is equiconsistent with the existence of a countably additive extension of Lebesgue measure that is defined on all sets of real numbers. Given such an extension, does there exist a set of real numbers whose symmetric difference with each Borel set of real numbers has positive measure?



Measure algebra of a total extension of Lebesgue measure
Yes. This follows from a theorem of Gitik and Shelah which says that the measure algebra of any such extension is nowhere separable. Hence by Maharam's theorem, there is some set which divides every borel set into two pieces of equal measure.


An immediate way to see this is as follows: Assume for some total extension $m$ of Leb. measure, this fails. Then $\mathcal{P}(\mathbb{R}) / Null(m) \cong Borel / Null(Leb.)$ which contradicts a theorem of Gitik and Shelah.


A "forcing-free" proof of Gitik-Shelah theorem can be found in: A. Kamburelis, A new proof of the Gitik-Shelah theorem, Israel Journal of Mathematics, October 1990, Volume 72, Issue 3, pp 373-380


