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June 28th, 2016, 10:29 AM  #1 
Newbie Joined: Jun 2016 From: India Posts: 3 Thanks: 0  Probability  Sigma algebra
I read in a book "The elements of $\displaystyle sigma $ algebra over a sample space $\displaystyle omega$ are called events" and not all subsets of a sample space are events. Can you please explain this definition?

June 28th, 2016, 02:47 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,397 Thanks: 546 
A (set) algebra is a collection of sets (events) which is closed under any finite combination of unions, intersects, and complements, which includes the entire set (sample space). A sigmaalgebra extends this notion to include countable combinations as well. Note that measure theory and probability theory are very similar, but the terminology differs: measure theory  sets, probability theory  events. 
June 28th, 2016, 11:47 PM  #3 
Newbie Joined: Jun 2016 From: India Posts: 3 Thanks: 0 
Actually I a newly studying the probability theory and would like to know why the definition of events is given as following "The elements of a sigma algebra over a sample omega are called events" and why not all subsets of the sample space of a particular experiment can be events?

June 29th, 2016, 04:00 AM  #4 
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16 
There are two reasons why one considers only set belonging to a sigmaalgebra instead of all subsets of Omega. 1) a suitable sigmaalgebra is all what is needed. Suppose Omega is the set of real numbers R and X is a real random variable. All the interesting questions you may ask about X is whether X<a for some real number a, or if X belongs to the interval [a,b] or (a,b), (a,b]... so that you only need to consider the sigmaalgebra generated by theses sets (which is much smaller than the power set of R) 2) you might nevertheless consider all subsets of Omega. However, if you do that, you will find that for an Omega big enough there are some probability measures which behave pathologically. This is related to the socalled BanachTarski paradox: in the 3dimensional euclidean space there is no rigidmotioninvariant measure on all subsets (but there is one on the borelian sigmaalgebra). 

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algebra, probability, probaility, sigma, sigma algebra 
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