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I am taking a course on real analysis (mainly about Lebesgue measure etc') and a few lectures back the lecture introduced to concept of algebra and $\sigma$-algebra.

It feels a bit strange to see it in this context first for me, is it studied in another area of mathematics and is just useful here, or does it studied in analysis and is mainly used in this context ?

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Not sure what to tag this...feel free to retag –  Belgi Nov 20 '12 at 16:39
    
The field where they appear in a general setting is usually called "measure theory". –  Antonio Vargas Nov 20 '12 at 16:47
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Real analysis and probability I guess, main places where those are studied –  Jean-Sébastien Nov 20 '12 at 16:47
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Algebras and $\sigma$-algebras of sets are particularly useful in analysis for studying spaces where you want to be able to perform "robust" actions like taking unions and intersections of the objects in your collection, sometimes of a countable subcollection. In my experience, it comes up a lot in a first course in measure theory where the fundamental object of a study -- a measure space, needs to be a $\sigma$-algebra, because ideally you want to talk about the measure of unions of sets, their intersection, etc. Otherwise, Boolean algebras and Heyting algebras are algebras used in logic... –  Isaac Solomon Nov 20 '12 at 16:48
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In probability theory, you want your "set of possible events" to be a $\sigma$-algebra. Why? You'll want to consider both the possibility of an event occurring or not occurring (i.e. this set should be closed under complimentation), and you want to be flexible in considering "whats the probability one of these many things happens?" (the set should be closed under unions). –  icurays1 Nov 20 '12 at 16:53

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Algebras and $\sigma$-algebras are just Boolean algebras. In set theory there are applications (and study) of Boolean algebras, although often we discuss complete algebras. Namely, the union and intersection of any collection, not just finite or countable collections (e.g. power sets).

In measure theory, and its "applicative branch" - probability theory, one uses $\sigma$-algebras to describe the collections of measurable events.

One can then take one step further into set theory where the study of Borel sets and projective sets (both $\sigma$-algebras) is important, and takes quite some of the attention.

Lastly, the theory of Boolean algebras have merits on its own and some surprising applications to other fields of mathematics. Reconstruction theorems of Rubin are such example (given two spaces that an automorphism group of both is isomorphic under some condition, can we deduce the spaces were isomorphic to begin with? Yes, sometimes).

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Thanks for the answer Asaf, I suspected it had something to do with set theory, but then I thought that without any additional structure we can't really do anything, nice to hear that I was wrong on that one :) –  Belgi Nov 20 '12 at 19:29

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