Universally measurable set

In mathematics, a subset ${\displaystyle A}$ of a Polish space ${\displaystyle X}$ is universally measurable if it is measurable with respect to every complete probability measure on ${\displaystyle X}$ that measures all Borel subsets of ${\displaystyle X}$. In particular, a universally measurable set of reals is necessarily Lebesgue measurable (see § Finiteness condition below).

Every analytic set is universally measurable. It follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective set is universally measurable.