TITLE: Non-measurable subset of $\omega_1$
QUESTION [9 upvotes]: Consider $\omega_1$ equipped with the order topology. Then Borel subsets of $\omega_1$ are precisely those which contain a closed and unbounded set or the complement contains such a set. There must be (in ZFC) sets which lack this property, as otherwise $\omega_1$ would be measurable. Can one please tell how to "construct" such sets (using some form of choice, of course).

REPLY [6 votes]: It requires the axiom of choice, but there are sets which are stationary and co-stationary. Namely they do not contain a club not they a disjoint from one. 
For example using Solovay's theorem we can partition $\omega_1$ into $\omega_1$ disjoint stationary sets. 
In some models where the axiom if choice fails every subset of $\omega_1$ contains a club or is disjoint from one.