Thread: lebesgue measure

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?

A counterexample: Cantor set is a set with measure 0 but uncountable.