TITLE: Why can't all subsets of sample space be considered as events?
QUESTION [7 upvotes]: 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?

REPLY [4 votes]: 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.
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.)
If you really want to know more about non-measurability, there are Wikipedia articles:
http://en.wikipedia.org/wiki/Non-measurable_set