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From earlier question here. Consider

$\mu \left( (0,1) \cap \mathbb Q \right) = 0$

where $(0,1) = \left(0,\frac{1}{3}\right)\cup \left[\frac{1}{3}, \frac{2}{3}\right] \cup \left(\frac{2}{3}, 1\right)$ so

$ \mu ((0,1) \cap \mathbb Q ) = \mu \left(\left(0,\frac{1}{3}\right) \cap \mathbb Q \right) + \mu\left( \left[\frac{1}{3}, \frac{2}{3}\right] \cap \mathbb Q\right) + \mu\left(\left(\frac{2}{3}, 1\right) \cap \mathbb Q\right) = 0 + \frac{1}{3} + 0 \neq 0$

contradiction with latter case. Why did my partition change the result?

[solved]

$\mu\left(\left[\frac{1}{3}, \frac{2}{3}\right] \cap \mathbb Q\right) = 0$.

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    [Continued] Terry Tao wrote a good introduction to "The Problem of Measure": http://terrytao.wordpress.com/2010/09/04/245a-prologue-the-problem-of-measure/2011-09-10

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