Example 7.3.5(b): Properties of Outer Measure
The set [0, 1] is not countable.We have already proved this fact by elementary means (trying a diagonalized counting process and arriving at a contradiction). With the advanced machinery now at our disposal, the proof is only a single line.
If [0, 1] was countable, m*([0, 1]) = 0. But m*([0, 1]) = 1 - 0 = 1, a contradiction.