I just finished a course where we discussed concepts such as Lebesgue integration and Lebesgue measure of sets. Today, I was telling my brother about how the irrationals on the interval [0,1] have measure 1, which is sort of counter-intuitive.(adsbygoogle = window.adsbygoogle || []).push({});

Anyway, he proposed the following question. Let A be the set of irrationals on the interval [0,1] whose first digit in their decimal expansion is even. What is the measure of A? Intuitively, I feel like it should have measure 1/2, since it should capture "half" of the irrationals on [0,1]. However, I can't think of any way to cover these irrationals with open intervals of any total length less than 1.

So if A does have measure 1/2, how can we prove that? If A has measure 1, then how do we reconcile this with the fact that the measure of the irrationals on [0,1] is 1?

Thanks!

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# Measure of Irrationals with Even First Digit

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