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June 28th, 2016, 09:29 AM   #1
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Smile 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?
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June 28th, 2016, 01:47 PM   #2
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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 sigma-algebra 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.
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June 28th, 2016, 10:47 PM   #3
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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?
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June 29th, 2016, 03:00 AM   #4
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There are two reasons why one considers only set belonging to a sigma-algebra instead of all subsets of Omega.

1) a suitable sigma-algebra 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 sigma-algebra 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 so-called Banach-Tarski paradox: in the 3-dimensional euclidean space there is no rigid-motion-invariant measure on all subsets (but there is one on the borelian sigma-algebra).
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