Let P be a closed set on a finite interval [a,b]

a) prove P is a Lebesgue measurable

b)Suppose its lebesgue measure m(P)=0. is it true that P is countable? Prove or give a counter example.

since borel set is measurable, i dont think it is that hard to prove a)

but i dont know how to do b). would someone help me on this please?