On the page 43 of Real Analysis by H.L. Royden (1st Edition) we read: "(Ideally) we should like $m$ (the measure function) to have the following properties:
- $m(E)$ is defined for each subset $E$ of real numbers.
- For an interval $I$, $m(I) = l(I)$ (the length of $I$).
- If $\{E_n\}$ is a sequence of disjoint sets (for which $m$ is defined), $m(\bigcup E_n)= \sum m (E_n)$."
Then at the end of page 44 we read : "If we assume the Continuum Hypothesis (that every non countable set of real numbers can be put in one to one correspondence with the set of all real numbers) then such a measure is impossible," and no more explanation was given.
Now assuming the Continuum Hypothesis I am not able to see why such a measure is not possible. Would you be kind enough to help me?