1
$\begingroup$

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 ?

  • 0
    as icurays1 said, $\sigma$-algebras are useful to reasonably define probability measures. Actually, $\sigma$-algebras allow us to define "minimal sets of possible events" on which a probability measure is defined, which is necessary because you can't use too big sets of possible events. For example, you cannot find a reasonable definition of the measure of any subset in R.2012-11-20

1 Answers 1

2

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).

  • 0
    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 :)2012-11-20