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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:

  1. $m(E)$ is defined for each subset $E$ of real numbers.
  2. For an interval $I$, $m(I) = l(I)$ (the length of $I$).
  3. 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?

  • 5
    Please do **not** write all-caps. You should edit this to fix it...2012-01-03
  • 6
    @Kamram: To explain, on the web, writing in all capitals [is considered shouting](http://en.wikipedia.org/wiki/All_caps#Computing). You are basically yelling at people.2012-01-03

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