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I always wondered why $\sigma$-field is necessary in probability theory. I read Asaf Karagila's answer here that nicely points out the significance of $\sigma$-field. But I couldn't conceive why it is essential. After all,I have the sample space; so can't I work with it alone?

I've read many books, wikipedia on this issue & everyone answered in order to create a mathematical construct to model real-world processes, $\sigma$-field is necessary. I confess I am not at such a level to conceive what mathematical construct actually is. Nevertheless, can't I model the real-world processes only with sample-space $S$? I'm really confused. Can anyone please explain me the reason?

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  • Because you must have measurable sets and you need to apply limit processes, such that the continuity from below/above of the probability. In addition, it is not always possible to work with the power set of the sample space. – DonQuixote Sep 20 '15 at 04:14
  • We want to discuss the probability of events, which are (certain) subsets of the sample space. With discrete random variables, we could get away with less formal machinery. – André Nicolas Sep 20 '15 at 04:14
  • @André Nicolas: Hi, sir; ... can't we work with power set of sample-space? Why need $\sigma$-field? How is the later different from the former?? –  Sep 20 '15 at 04:16
  • For the point said by @AndréNicolas I recommend you the first chapter of Sinai's book (Theory of probability and random processes). – DonQuixote Sep 20 '15 at 04:17
  • The full power set will often not support a probability measure. – André Nicolas Sep 20 '15 at 04:19
  • The $\sigma-$additivity of $P$ fails, in general. – DonQuixote Sep 20 '15 at 04:20
  • Let me try again on @AndréNicolas's points. There are power sets so 'large' that it is not possible to assign probabilities to all without contradictions.(Not obvious, not convenient, but true.) The power class of the real line is an example. No problem using the power class if you have a sample space with a finite number of elements. – BruceET Sep 20 '15 at 07:09

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