A closely related question: I am reading a thesis that claims that the property "All measures on metric spaces are $\tau$-smooth" is independent of ZFC. A measure on Borel sets is $\tau$-smooth iff for any directed family of opens $U_t$, we have $\mu(\bigcup U_t) = \sup_t \mu(U_t)$. Robin Cooper's answer above tells us why the claim is plausible (non-inner regular measures exist when measurable cardinals do), but the author does not give a usable citation (only an out of print textbook, instead of a primary source). Does anyone have a pointer into the literature.

A closely related question: I am reading a thesis that claims that the property "All measures on metric spaces are $\tau$-smooth" is independent of ZFC. A measure on Borel sets is $\tau$-smooth iff for any directed family of opens $U_t$, we have $\mu(\bigcup U_t) = \sup_t \mu(U_t)$. Robin Chapman's answer above tells us why the claim is plausible (non-inner regular measures exist when measurable cardinals do), but the author does not give a usable citation (only an out of print textbook, instead of a primary source). Does anyone have a pointer into the literature.