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My textbook is Probability and random processes by Grimmett & Stirzaker and the first chapter does not explain this," for reasons beyond the scope of the book". The authors introduce the reader to sample spaces and to events and then go on to say that events are subsets of sample space. Then they ask, "Need all subsets of sample space be events ?" and then they say no. But I don't see why not. Can anyone give mean an intuitive explanation for this?

Did
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Anupam Kumar
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    Events in probability spaces (Wikipedia) – Zev Chonoles Sep 06 '13 at 04:29
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    It would help if you can give us some context. Also, do you really mean "all elements" or "all subsets"? – Nate Eldredge Sep 06 '13 at 04:33
  • I agree with Nate. Are you familiar with the concept of a sigma algebra? – Jemmy Sep 06 '13 at 04:37
  • "All subsets". Sorry about the confusion. The Context - The first chapter of the textbook - Probability and random processes - Grimmett & Stirzaker. They introduce the reader to sample space and to events and then go on to say that events are subsets of sample space. Then they ask, "Need all subsets of sample space be events ?" and then they say no. I don't see why not. I can see that in certain cases the answer is no, like the event where a coin comes out as both heads and tails. But, that does not mean "no" all the time right ? – Anupam Kumar Sep 06 '13 at 04:46
  • For infinite sets, whether this is possible depends on the size of the space, and is independent of the usual axioms of set theory. This is the Banach-Ulam measure problem, intimately tied up with large cardinals. See section 1A here. – Andrés E. Caicedo Sep 06 '13 at 06:20

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If the sample space is finite, then it's perfectly fine to take every subset of the sample space as an event.

If the sample space is infinite, you can sometimes run into contradictions if you try to assign a probability to every single subset of the sample space, essentially because infinite sets have a lot of subsets. [1]

This issue really only applies to subsets that can't be uniquely described in finitely many steps, though. (More precisely, if you don't assume the axiom of choice, there are models of the reals where this doesn't happen and you can take every subset of the sample space to be an event. [2])

If you really want to know more about non-measurability, there are Wikipedia articles:

http://en.wikipedia.org/wiki/Non-measurable_set

[1] Sadly, while this statement means something to people who know what it means, it's probably totally unhelpful to people who would need to ask this question in the first place.

[2] If you don't already know what this means, it's very unlikely that learning what it means would in any way help your understanding of probability. You're talking about probably a several-months detour to learn axiomatic set theory, learn about all sorts of crazy things you never imagined, and then ultimately learn that they're all basically totally useless.

Daniel McLaury
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google vitali sets......you cannot measure all the sets if you assume axiom of choice to be correct...however recently there was a paper which showed that you can measure almost all sets if you do not assume axiom of choice

user24367
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  • Assuming we're talking about Lebesgue measure, it's consistent with ZF that all sets are measurable: http://en.wikipedia.org/wiki/Solovay_model – Anthony Carapetis Sep 06 '13 at 04:37
  • yeah thats what I was saying.....but here it reads out "Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC" --- I am sure you will give rise to several other paradoxes......I am fine as long as assuming axiom of choice doesnt affect me in other ways....remember Hahn banach theorem – user24367 Sep 06 '13 at 04:43