For the first question, remember the Bernstein set S, which is the set of one representative of each equivalence classes, here the equivalence relation ~ is defined as x ~ y iff x-y is rational. Index the rationals in [0,1] by {r_n}, and let S_n=S+{r_n}. Then check that each S_n is disjoint from others, union S_n is contained in [-1,2], i.e. m(union S_n)=m*S for all n, i.e. lim m*E_n>=m*S>0. This is the counterexample._________________Few, but ripe.

—- Carl Friedrich Gauss