Background/Motivation. We know that some of the usefulness of Martin's Axiom lies in giving certain "smallness" properties to sets of size less than continuum, e.g. we have that for all infinite cardinals $ \lambda < 2^{\aleph_0}, 2^{\lambda} = 2^{\aleph_0}$; also all sets of reals of cardinality less than $2^{\aleph_0}$ are Lebesgue measure zero sets.
But I note at the end of the article "Between Martin's Axiom and Souslin's Hypothesis" (by K. Kunen and F. Tall) that the topological characterization of MA restricted to compact hereditarily separable spaces is consistent with having $\aleph_1 < 2^{\aleph_0} < 2^{\aleph_1}$. Which leads me to wonder...
Question: Is there any known restriction/modification of MA ( + not-CH, of course) which is consistent with existence of a non-Lebesgue measurable set of reals of cardinality less than $2^{\aleph_0}$?
(The proof of small sets being Lebesgue null (assuming MA) in Jech's set theory book seems so simple and compelling, it would make an affirmative answer rather intriguing.)