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Why are sigma fields significant in probability theory? 
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#1
May3011, 01:47 AM

P: 59

As the title.
Why are sigma fields important in probability? The only one reason I can think of is that sigma fields are used as domain, e.g. borel fields uses sigma fields instead of power set. However, are there any other significances of sigma fields in probability theory? Thanks for your response. 


#2
May3011, 03:17 AM

P: 68

Well, the probability, as any other measure, is defined over sigmafields primarily because of the existence of sets which are unmeasurable. We need to exclude those somehow, so this is the best way to do it.
As far as I know, all those unmeasurable sets are pretty pathological in their nature, and any subset that you can think of is most likely measurable. So, from a practical POV, this is just a formality which needed to be done for consistency's sake. I'm sure someone will correct me if I'm wrong. 


#3
May3011, 03:15 PM

Sci Advisor
P: 6,071

Probabilities can be added as long as the number of terms is countable. Sigma fields insure that if you have a countable number of events the union is also an event, so the calculation of its probability is meaningful.



#4
May3011, 04:14 PM

Sci Advisor
P: 1,807

Why are sigma fields significant in probability theory?
For general sigmaalgebras with corresponding measures, we can approximate by disjoint unions of some generating semiring. We could use the boxes in the lebesguemeasure case since the set of boxes is a generating semiring for the set of lebesguemeasurable sets. If we do not know that the set is contained in our sigmaalgebra (that is, being measurable), we can't use this method. In fact, sigmaalgebras have no use in consistency. Not using them, and still applying a measure to some set would be meaningless as we would have no method of calculating it. Applying a measure does mean that we have such a method (and generating semirings provide in many cases a constructive method!). We could try by e.g. approximating with some arbitrary methodsuch as by disjoint union of boxesbut we don't know that this will converge uniquely. And this is exactly what the existence+uniqueness of measures proves for sigmaalgebras. Similarly for probability theory; P(A) does not makes sense unless A is contained in a set where the measure P can be meaningfully applied. 


#5
May3011, 10:27 PM

P: 523

Further to the above comments, the use of sigma fields simply puts probability theory on a common foundation with measure theory, i.e. general integration.
It is possible to define versions of probability theory based on finite additivity rather than sigmaadditivity, with various implications, for example densities of sets on the natural numbers such as [itex]d(A)=\lim_{n\to\infty}\frac{1}{n}\#(A\cap\{1,...,n\})[/itex] are finitely additive but not sigma additive because all singletons have zero density. 


#6
Jun211, 03:00 AM

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P: 1,807




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