I am wondering whether it is possible to translate the idea of drawing a number randomly from the set of all natural numbers. If we have infinite additivity as an axiom this obviously does not work. What I wonder however is whether giving up infinite additivity and only insisting on finite additivity allows us to work with this kind of situations. And furthermore I wonder how useful it would be to work within an axiomatization without infinite additivity.

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    $\begingroup$ Even with just finite additivity, what probability would a uniform distribution necessarily assign to each natural number? $\endgroup$ – whuber Apr 18 '18 at 15:49
  • $\begingroup$ the probability would of course be 0. $\endgroup$ – Sebastian Apr 18 '18 at 15:56
  • $\begingroup$ It seems that answers your question. $\endgroup$ – whuber Apr 18 '18 at 16:10
  • $\begingroup$ If the only events of interest are infinite sets of numbers (like even or odd) this can be useful, and maybe used in number theory. But you can never simulate from such a distribution, and never use with data, so far as data consists of individual numbers $\endgroup$ – kjetil b halvorsen Apr 19 '18 at 0:35

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