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In many introductory courses to measure theoretic probability, the author usually introduces a few somewhat pathological examples on why a measure theoretic approach is needed. I am wondering if there are straightforward and applicable examples that can highlight why the undergraduate version of probability fails.

user321627
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    What do you mean why the undergraduate version fails? The undergraduate version is based on measure theory, they just don't talk about it, and instead focus on computation. – Andrew Feb 02 '23 at 02:58
  • For example, what does i.i.d random variables mean to you? – Andrew Feb 02 '23 at 03:01
  • @AndrewZhang For example, the undergraduate version would say $P(X\mid Z) = P(X,Z)/P(Z)$ but if $Z$ is continuous this fails to be rigorous. I am wondering whether there is a very applicable example which can demonstrate why measure theory is necessary. – user321627 Feb 02 '23 at 03:06
  • Even using the abstract definition of conditional expectation from measure theory, one cannot cohesively condition on measure zero sets. See Grimmett and Stirzaker Probability and Random Processes 3e section 4.7 for an example. – Andrew Feb 02 '23 at 03:15
  • Possibly useful: David Thomas Price, Countable additivity for probability measures, American Mathematical Monthly 81 #8 (October 1974), pp. 886-889. – Dave L. Renfro Feb 02 '23 at 09:21
  • i think this is a duplicate but for the wrong thing it should be this https://math.stackexchange.com/questions/1506416/probability-and-measure-theory/1530321#1530321 currently the question that this is supposedly a duplicate of is asking about generalised measure theory outside of probability – BCLC May 19 '24 at 11:39

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