7.3. Measures

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.

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