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