Is there a video that provides intuition about why countable subsets of R (even dense ones) have outer measure zero, by zooming in on a cover of such a set by a union of countably many intervals?
I remember being very confused as a child about why a union of open intervals containing all the rational numbers wasn't necessarily the whole real line.
@johncarlosbaez@JimPropp I've mentioned here before that I find this the most counterintuitive thing I know in mathematics - even though it's sort of obvious.
@dpiponi@johncarlosbaez@JimPropp Isn't it a fractal thing? If you consider rationals with denominators less than some n, you can cover them with open intervals small enough to leave big gaps between them. Then zoom on one such gap. If you see there rationals with denominators less than n+1, cover them with smaller open intervals, again leaving gaps, and so on. You can always make gaps larger than the current cover. I think this works, doesn't it?
@BartoszMilewski - yes, this works. Another trick is this: enumerate all the rationals and cover the nth rational with an open interval of length 2⁻ⁿ ε. Then we've covered all the rationals with an open set of measure at most ε. So this is an open dense set S(ε) of arbitrarily small measure.
A further spinoff: take the set S(1/2) ∩ S(1/3) ∩ .... This has measure zero, but it contains a lot of numbers that aren't rational. It's uncountable! In fact it's 'comeager', which is a bigness property of subsets of ℝ that's stronger than being uncountable.
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