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Let $I$ be the set of all irrational points in $[0,1]$ and $\{J_n\}_1^N$ be a finite system of open intervals that cover $I$ . How to show that the $\sum_1^N \operatorname{length}(J_n) $ is greater or equal $1$.

Does it remain true if it is countable system of intervals?

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    The rationals are a null set, so $I$ has measure $1$.2012-09-05

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