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jimrandomh comments on Open Thread: July 2010, Part 2 - Less Wrong
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Any irrational number drawn from what distribution? There are plenty of distributions that you could draw irrational numbers from which do not have this property, and which contain the same number of numbers in them. For example, the set of all irrational numbers in which every other digit is zero has the same cardinality as the set of all irrational numbers.
Yes, although generally when asking these sorts of questions one looks at the standard Lebesque measure on [0,1] or [0,1) since that's easier to normalize. I've been told that this result also holds for any bell-curve distribution centered at 0 but I haven't seen a proof of that and it isn't at all obvious to me how to construct one.
Well, the quick way is to note that the bell-curve measure is absolutely continuous with respect to Lebesgue measure, as is any other measure given by an integrable distribution function on the real line. (If you want, you can do this by hand as well, comparing the probability of a small bounded open set in the bell curve distribution with its Lebesgue measure, taking limits, and then removing the condition of boundedness.)
Excellent, yes that does work. Thanks very much!