I would actually want to use only one limit, not two independent limits. And if I throw this [1] at Wolfram Alpha it actually says 1. I have a set of n elements

I would actually want to use only one limit, not two independent limits. And if I throw this [1] at Wolfram Alpha it actually says 1. I have a set of n elements with measure 1/n and then grow the set towards infinity while simultaneously shrinking the measure. I agree that this does not work if growing the set and shrinking the measures are two independent limiting processes very similar to how integrating x dx from minus to plus infinity yields infinity if you have independent integration limits [2] but yields 0 if the integration limits are not independent [3]. I am still happy to accept that it requires uncountable sets but I am not convinced by the argument you provided, that the limit does not work out. I think there must be a different issue, some other property of probability measures that fails. EDIT: I also finally did a little bit of searching and while I did not read much yet, it seems that the problems indeed arise from additivity as you hinted at with the partial sums. But I also found that there are actually ways to have uniform distributions on the natural number [4] if one uses non-standard axioms, but I only skimmed the paper for the moment. [1] https://www.wolframalpha.com/input/?i=lim_%28n-%3E%E2%88%9E%29%28+sum_%28i%3D0%29%5E%28n+-+1%29+1%2Fn%29 https://www.wolframalpha.com/input/?i=lim_%28n-%3E%E2%88%9E%... [2] https://www.wolframalpha.com/input/?i=lim_%28a-%3E-%E2%88%9E%29%28lim_%28b-%3E%2B%E2%88%9E%29+integral_a%5Eb+x+dx%29 https://www.wolframalpha.com/input/?i=lim_%28a-%3E-%E2%88%9E... [3] https://www.wolframalpha.com/input/?i=lim_%28a-%3E%E2%88%9E%29+integral_%28-a%29%5E%28%2Ba%29+x+dx https://www.wolframalpha.com/input/?i=lim_%28a-%3E%E2%88%9E%... [3] http://cetus.stat.cmu.edu/tr/tr814/tr814.pdf http://cetus.stat.cmu.edu/tr/tr814/tr814.pdf