So I know that the Lebesgue outer measure of a set of only countably many points is 0. An example of this is the rationals as a subset of the reals.(adsbygoogle = window.adsbygoogle || []).push({});

I want to make sure my intuition behind this is correct. The process: Now, if we are going to take the Lebesgue outer measure of the rationals, we first want to take every open set covering them, then for each such cover sum together the length of the individual intervals making up the covering, and finally if we construct a set representing the sums of these individual coverings, if we take the inf of this set that is the Lebesgue outer measure.

The intuition (as I think I understand it) So let's consider the rational numbers as a subset of the reals. First thing to notice is that this subset is just a bunch of (countably infinite) disjoint points. (The points are disjoint because between any two rationals, you can find infinitely many irrationals, and there are countably many because the cardinality of the rationals is aleph-0). So say we want to begin by constructing any arbitrary open covering of the rationals, we can do this by picking each rational q, then constructing an open set of the form (q-ε, q + ε) for some arbitrary ε>0. So our covering consists of countably many intervals like this. For our covering, the "sum" of the lengths of these countably many intervals will then be Ʃ(q+ε-(q-ε)) = Ʃ(2ε) for all the rationals q. So to find the Lebesgue outer measure, we form this set made up of all such sums, one for each individual cover, and look at the inf of that set. It seems like the inf will be the number we get as we take ε smaller and smaller, arbitrarily close to 0 (since the individual intervals in the cover could get arbitrarily small in length since we're just covering a bunch of disconnected points). So this number would just be 0. It seems intuitive that we could generalize this beyond the Rationals example, and to any subset of countably many points because you could perform a similar process on any such subset.

Does that sound even vaguely right? I'm just trying to wrap my mind around the basics of Lebesgue integral and measure theory and Lebesgue measure and want to make sure I'm on the right track.

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# Lebesgue outer measure of a set of countably many points is 0 - logic check

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