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In Durrett’s Probability: Theory and Examples I have found the following proof for the „Martingale convergence theorem“:

from Durrett p. 202, ed. 4.1 (online version)

My question is about the first part where he says „Since the last conclusion holds for all rational a and b, …“. Why does it hold for all rationals? This probably has some measure theoretic reason which I couldn't figure out.

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  • $\begingroup$ The statement holds for any real numbers and hence in particular for all rationals. Where exactly is your problem with that...? $\endgroup$ – saz Feb 1 '18 at 17:40
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    $\begingroup$ If you're wondering why he considers only rationals, as opposed to all reals, there is a measure-theoretic reason: it's so that the union is countable. $\endgroup$ – grndl Feb 1 '18 at 17:44
  • $\begingroup$ @aduh: Why do we want the union to be countable? $\endgroup$ – Quasar Feb 1 '18 at 19:31
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    $\begingroup$ A countable union of null events is null. An uncountable union of null events could have positive probability. $\endgroup$ – grndl Feb 1 '18 at 21:13

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