As far as I know, this is quite a tricky result. There is a proof of it in Halmos's Measure Theory (Theorem E, p.70).

Given a set M of positive measure, it's a very natural idea to think that would provide an answer to the problem, but unfortunately that won't work. The difficulty is that a general non-measurable set P might be concentrated in one half of the interval, and M in the other half. The standard construction of a non-measurable set P does appear to give a set that is somehow spread right through the interval, but Halmos's proof requires a strengthening of that construction.