[Math] the interval $[0,1]$ without the rational numbers is not a countable set

Question

I have to prove that the interval $[0,1]$ without the rational numbers is not a countable set, but I have to prove it using Lebesgue measure theory! Could anybody help?

Answer 1

Well, if you put $\;A:=\Bbb Q\cap[0,1]\;$ , then $\;m(A)=0\;$ since $\;|A|\le|\Bbb Q|=\aleph_0\;$ , so

$$m([0,1]\setminus A)=m([1,0)]-m(A)=1$$

and thus $\;[0,1]\setminus A\;$ cannot be countable.