**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.)