I am not a mathematician, rather I pick up on topics on the go, when I need something for the topic I am studying in the given time. So I am sorry if this is trivial to most of you and apologies for any conceptual mistakes I might make in the description - I'll try to be as precise as possible.
At the moment, I am studying Probability Theory, from this course: https://www.youtube.com/playlist?list=PL5B3KLQNAC5jT6yjV1199ji1zUy1YUp6P , [for the purposes of understanding Stochastic Calculus for Finance (Vol. II - S. Shreve)], and I stumbled upon sigma algebras.
While I do understand the concept; if we have a set which is a collection of subsets of Omega (i.e. if we have a collection of events) denoted by F, then F is a sigma-algebra if it satisfies the following three conditions;
- Omega belongs in F,
- F is closed under complements,
- F is closed under countable Unions
So far so good and I also understand the properties that derive from the definition as well as how they are derived. In addition I know that we have the trivial sigma algebra, the smallest sigma algebra on Omega and the Discrete Sigma Algebra, which is the power set of Omega, being the largest sigma algebra on Omega.
My problem is with generated sigma algebras. I do understand the definition; Let A be an arbitrary collection of subsets of Omega, then sigma(A) is the generated sigma algebra, generated from A and is the smallest sigma algebra containing A. Further, we can find the smallest sigma algebra by intersecting all sigma algebras containing A, as the intersection of sigma algebras is also a sigma algebra.
The last part is the one I don't understand and confuses me. I get that we have the power set of Omega that definitely contains the collection A - But what exactly do we mean by intersecting all the sigma algebras containing A to find the smallest one containing A? Does it mean that if we have a sigma algebra containing the collection A and another collection of subsets, B (which is a sigma algebra containing A, but i get that it is not the smallest) and intersect it with the power set of Omega, we generate sigma(A), which is indeed the smallest and more refined to answer the questions that we need in our problem? But, where exactly does the bigger sigma algebra (on the collections A and B) come from?
If anyone could provide a more intuitive explanation or even better give an example (finite, like a die roll), I would be very grateful.
Many thanks for your time in reading this! :)