TITLE: Relation between projective hierarchy and universally measurable sets
QUESTION [8 upvotes]: Let $X$ be Polish. It is known that every analytic and coanalytic subset of $X$ is universally measurable. The Wikipedia article about universally measurable sets notes that (assuming projective determinance) every set in the projective hierarchy is universally measurable. Are there also universally measurable sets that are not related to the projective hierarchy, i.e. can not be constructed in this way?

REPLY [10 votes]: Thanks, Joel, for mentioning our paper. 
By Martin-Steel, if there exist infinitely many Woodin cardinals, then every uncountable projective set contains a perfect set, so, by the result of Hausdorff mentioned by Joel there are universally null sets which are not projective. 
On the other hand, it is apparently an open question whether it is consistent for all universally measurable sets to be $\Delta^{1}_{2}$ (see problem CG on Fremlin's list : https://www.essex.ac.uk/maths/people/fremlin/problems.pdf). 
Finally, I should mention two improvements of Hausdorff's theorem : 
(Reclaw) Any set of reals which is wellordered by a universally measurable relation is universally null (https://projecteuclid.org/download/pdf_1/euclid.rae/1212763971)
(Grzegorek) There is a universally null set whose cardinality is the smallest cardinality of a non-Lebesgue-null set. A proof of this can be found in Bukovsky's book (http://www.springer.com/gp/book/9783034800051)