Discussion
Which of the following subsets of \(\mathbf{R}\) has non-zero Lebesgue measure ?
Answer: D
The set with non-zero measure is the set of limit points of \(\mathbf{Q}\) in \(\mathbf{R}\).
Every real number is a limit of rationals, so the set of limit points of \(\mathbf{Q}\) is all of \(\mathbf{R}\), which has infinite Lebesgue measure.
The others have measure 0. \(\{n + 1/n\}\) and \(\{a + b\sqrt{2} : a, b \in \mathbf{Q}\}\) are countable, and countable sets are null. The Cantor set is uncountable, but it is left after removing intervals of total length \(\sum 2^{n-1}/3^{n} = 1\) from [0, 1], so its measure is 0.
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