# Measurable and definable sets

Is there a model of set theory such that:

1. AC holds,

2. Every ordinal definable set is measurable,

3. Every ordinal definable set of sets of $\mathbb{R}^2$ whose projection on the first (or second axis) is ordinal definable contains a definable member.

• In statement 3, you say "set of sets of $\mathbb{R}^2$, but I took this to be just about a "subset of $\mathbb{R}^2$". Also, it should be stated that the set is non-empty to avoid a trivial contradiction. – Joel David Hamkins Oct 21 '13 at 1:34

• @Andres, I agree. But from ZF plus statement 3 we get a well-ordering of the reals, and perhaps this enough choice to define Lebesgue measure and prove the basic properties? And further, we have $\mathbb{R}\subset\text{HOD}$, which would give rise to the $\text{HOD}$ Lebesgue measure, as ZFC holds there. – Joel David Hamkins Oct 21 '13 at 1:05
• That is enough choice to get its basic properties, since it gives CC($\mathbb{R}$). $\;$ – user5810 Oct 21 '13 at 1:30