The usual proof of the non-measurable set is to prove that it has outer measure 1 and inner measure 0. I like this proof, which contains the result you are looking for.
We know there exists a non-measurable subset in [0,1). Call it P.
P is a non-measurable set constructed by identifying the interval [0,1)
with the unit circle in R^2. If I can find it online, I'll post a link.
Let A be a measurable subset of P. Show that A has (Lebesgue) measure 0.
I must admit I'm stuck as to how to proceed