# Measure algebra of a total extension of Lebesgue measure

Solovay shows 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?

• A related question: mathoverflow.net/questions/204409/… – Ashutosh Jun 14 '15 at 16:52
• Interestingly, if you are willing to ditch the axiom of choice, you can get this extension without large cardinals! (Also due to Solovay) – Asaf Karagila Jun 14 '15 at 17:29

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.