Is it consistent with ZFC that there is a subset of $[0,1]$ whose cardinality is less than that of the continuum but which has positive Lebesgue measure?


Obviously not given CH. And, given ZFC, there is such a subset iff there is a subset of full measure that has cardinality less than that of the continuum. Moreover, I think it follows from the consistency of ZFC with the non-existence of Sierpinski subsets of $[0,1]^2$ that it is consistent with ZFC that there is a subset of $[0,1]$ whose cardinality is less than that of the continuum and which has positive *outer* Lebesgue measure.



A set of positive measure with cardinality less than that of the continuum?
No. It is a famous exercise that if $X\subset\mathbf{R}$ has positive measure then $X-X$ contains an interval. It follows that $X$ has cardinality continuum.


