A question about nonmeasurable subset of real line and uncountable measure-zero subset. When I was reviewing real analysis, I have a question below.

Does every nonmeasurable subset of real line have an uncountable and measure-zero subset?

Just consider the Lebesgue measure.
I have known that this is true for every measurable set whose measure is greater than zero.
 A: This cannot be answered by the usual axioms of set theory.


*

*It's consistent with ZFC that every set of reals of size $\aleph_1$ has measure zero (note that this requires $2^{\aleph_0}>\aleph_1$). For example, this follows from MA$_{\aleph_1}$. 
In this case  if $A$ is non-measurable, just consider any subset of $A$ of cardinality $\aleph_1$.

*Meanwhile, a consistent negative answer is easy to construct via transfinite recursion assuming CH (these are the Sierpinski sets). The key point here is that, while there are $2^{2^{\aleph_0}}$-many null sets in general, we can find a set $\mathcal{G}$ of $2^{\aleph_0}$-many (hence by CH, $\aleph_1$-many) null sets such that every null set is contained in one in $\mathcal{G}$, and this lets us set up a length-$\omega_1$  recursion (length-$\omega_1$ is useful since it means that at each step we've only "used countably many points" so there are lots of points "still untouched" as we continue to build our set).
Finally, I don't know what happens if both CH and MA$_{\aleph_1}$ fail.

As an aside, note the crucial role in the first bulletpoint of the number $$\mathfrak{m}=\mbox{the least size of a non-measurable set}.$$ This "$2^{\aleph_0}$-like" number is a cardinal characteristic of the continuum - there's a lot of interesting mathematics around what we can (and can't!) prove about CCCs.
